Category: Physics Notes PPT

Water is an essential part of this planet. It covers almost 71% of the Earth’s surface, of which 95% water is salty that is undrinkable. Only 2% of the water present is fresh and drinkable. Water has a wide range of uses. It is used for the survival of human beings, flora, and fauna. The importance of water can be estimated from the fact that 75% of the human brain is made up of water. Water is not only used for recreations and day-to-day activities but is also used in industries, agriculture, etc.

Chemical Composition of Water

The chemical composition of water is H₂O, which means one molecule of water is made of one oxygen atom and two hydrogen atoms held together by a strong covalent bond. Water is the lightest in gas form, whereas a liquid is much heavier than its solid form. It is a good conductor of electricity and has different properties. Water is colourless, odourless, and tasteless. Apart from these, there are various other properties that make water unique. Water is found everywhere in different forms. At poles, it is found in the form of ice, whereas at other places, it is found in the form of liquid.

Why is Water Unique?

Water is unique when it comes to its physical properties, yet when further research is done on this substance, it seems that even more unusual properties are revealed, like its anomalous behavior, especially when there’s ice involved. The anomalous expansion of water (hereafter referred to as AEW) was first discovered by British scientist W.J.M. Rankine in the year 1859. 

He found that when water is frozen, it occupies more space than when it is in its liquid form. This expansion can be as much as 9% at 0 degrees Celsius (32 degrees Fahrenheit), compared to just 1% for most other substances. The cause of this anomaly is still unknown today, though many hypotheses have been put forward over the years.

One theory suggests that the increased volume is due to the formation of tiny ice crystals; another proposes that water molecules from a more ordered structure, when frozen, take up more space in the process. However, neither of these theories has been proven conclusively.

AEW can have a significant impact on our lives, particularly in the field of architecture. In cold climates, for example, builders have to take AEW into account when designing structures; if they don’t, the buildings may be structurally unsound and could even collapse in severe cases. In addition, AEW can have an impact on the environment. For example, when water expands as it turns to ice, it can put pressure on the banks of rivers and lakes, leading to flooding. In addition, AEW can have an impact on the environment. For example, when water expands as it turns to ice, it can put pressure on the banks of rivers and lakes, leading to flooding. 

Despite its many quirks, water is still an essential part of our lives. It’s important to understand the properties of water in order to make the most efficient use of this vital resource. 

Properties of Water:

1. Amphoteric Nature: It can behave as both – acid and base.

2. Hydrolysis: Water has a very high dielectric constant.

3. Viscosity: It has high viscosity due to the intermolecular force of attraction.

4. Redox Reactions: Water is a great source to obtain dihydrogen.

5. Hydrates Formation: It is defined as the attachment of water molecules with any compound. There are various types of hydrates. They are:

• Coordination hydrates

• Interstitial hydrates

• Hydrogen bond hydrates

There are two properties of water due to the hydrogen bond. They are:

1. Cohesion: Molecules are close to each other because of the collective action of the hydrogen bond. Cohesive forces are responsible for the phenomenon of surface tension between water molecules.

2. Adhesion: It is the bonding of water molecules with other molecules. This characteristic causes capillary action, i.e., the ability of water to flow against gravity in a narrow space.

Anomalous Expansion of Water:

Water shows unnatural behavior, which makes it unique. Between a certain range of temperatures, it shows the anomalous expansion, which has many applications. This helps in the survival of marine life, which is one of the major importance of this behavior of water, in addition to many applications that are discussed.

From the below graph it can be seen.

Hence, it is clear from the graph that density increases as temperature rises from 4 degrees Celsius, and then density decreases after.

Why Does it Happen?

The water molecule is made up of one oxygen and two hydrogen molecules. The water molecules are held together because of the intermolecular force of attraction between them at normal temperatures. The molecules of water in liquid form are always unstable, moving zig-zag inside the container, constantly rearranging themselves.

On cooling the water, zig-zag motion decreases as the molecules lose their energy. On a freezing, the water molecules start squeezing, and water reaches its maximum density. This is caused by the hydrogen bonding between the oxygen atoms that are negatively charged and the hydrogen atom that is positively charged from two different molecules of water. 

The rise in temperature  will cause the ice to form water, not because of the O-O bond but due to H-O bond attraction. The lattice structure of ice prevents the movement of water molecules. But since the H-O bond is not as strong as the O-O bond, it will expand, and hence water will occupy more space. This can be understood with an example: Suppose 5 people are sitting on a seat with their hands folded, but as they start stretching their hands, there is a possibility that the corner one may fall. So, the same goes in the case of water also, as the H-O bond overtakes, it requires more space.

Applications:

This property of anomalous expansion of water has many applications.

• Aquatic Life: This unique property of water helps aquatic life to survive. With a further drop in temperature, water on top forms ice and becomes a bad conductor of heat. This stops the heat from escaping the water body and helps aquatic life to survive. Hence, aquatic life can survive even when the temperature reaches or falls below.

• Bottle Burst: If you put a fully-filled water bottle in the refrigerator and its temperature is below 4 degrees Celsius, then according to the anomalous behavior of water, the water inside t
  he bottle will expand. Therefore, due to no space in the bottle for the expansion of water molecules, water will exert force on the walls of the bottle and it will burst.

• Soft Drinks Bottle: Soft drink bottles are kept in freezers. So, as the temperature drops below 4 degrees Celsius, the drink inside the bottle starts expanding. So, to prevent the molecules from exerting force on the wall and bursting, there is a little space left inside the bottle.

• Pipe Burst: In cold countries, water pipes expand due to the anomalous behavior of water, and hence the water exerts a large amount of force on the pipe wall, and it bursts.

• Breaking of Rocks: Rocks break during winter because as the temperature decreases below, water inside the rocks starts expanding and hence exert a large amount of force on the rocks and results in the breaking of rocks.

We know that when a body is subjected to an external force, it starts accelerating and this is what Newton’s second law says. Acceleration is nothing but the changing velocity of an object in a unit of time. It is a vector quantity that bears both magnitude and direction. It is measured in ms-2;however, one more term lies in Physics and that is Radial Acceleration.

 

So, do you know what radial acceleration is? Well! When angular velocity changes in a unit of time, it is a radial acceleration. 

 

We know that a body can execute two types of motion and they are linear and circular motion. When it is linear motion, we consider just acceleration; however, during circular motion, which is actually an angular acceleration. 

 

We know that in a circular motion, the direction of the angular rate of velocity changes with time constantly so that’s why its angular acceleration gets two following components namely:

• Radial acceleration

• Tangential acceleration

So, let’s start with radial acceleration:

 

Radial acceleration is symbolized as ‘ar’ and it is the rate of change of angular velocity whose direction is towards the center about whose circumference, the body moves.

 

It happens because of the centripetal force. So centripetal force is the reason for a radial acceleration. 

 

A body whose mass is ‘m’ and the force acting on it is ‘mar’…..(1)

 

The formula for the centripetal force acting on the stone moving in a circular motion is mv2 /r….(2)

 

Equating (1) and (2):

mar = mv2 /r

 

So, we get the radial acceleration formula as:

ar = v2/r….(3)

 

Equation (3) is called centripetal acceleration. 

 

Radial Acceleration Units

The units of measurement are as follows:

Symbolically, these two units are written as ωs-2 or ms-2,respectively.

 

Radial Component

Let’s suppose that your child is on a merry-go-round. The direction of the velocity vector taken from your position will be tangential to the circular path in which the merry-go-round is making rounds. However, the centripetal acceleration points radially inwards or towards the center, which is what makes you go round. 

 

And from the formula in equation (3), we can see that the greater the radius of the circle of rotation, the lesser is its rate of change of velocity or the radial acceleration and vice-versa. Because of this reason we see that the smaller merry-go-rounds rotate a lot faster than the big ones. 

 

Now, it is crystal clear that the radial component is the primary reason for any object to keep making a circular motion.  

 

A body whose acceleration is always directed along the radius as its name signifies, there is one more component of acceleration whenever an object travels with a non-uniform speed and that is tangential acceleration ( at). The tangential acceleration acts tangentially to the path along which the object moves during a circular motion.

 

The below images show the variation of radial acceleration with the tangential acceleration:

 

 

This is for the centripetal or radial acceleration.

 

 

You can see the tangents drawn to the path of the object with the changing velocity.

 

We must keep in mind that the resultant acceleration is the sum of these two types of accelerations and the formula along with the required figure is stated below:

 

Formula: a = ar + at….(4)

 

Equation (4) states that the radial component of acceleration means the component of resultant acceleration which is perpendicular to the instantaneous velocity for the motion along any general path (not necessarily for circular motion). Since this component of acceleration is always directed along the radius of curvature of the trajectory (projectile motion), that’s why the name radial acceleration is given to this type of acceleration.

 

 

Radial And Tangential Acceleration 

• Radial acceleration is always along normal to the instantaneous velocity so it is also known as normal acceleration.

• Radial acceleration is always directed towards the instantaneous center of curvature of the trajectory so it is also named centripetal acceleration.

• Radial or centripetal acceleration is never defined only for circular motion, it may be defined for any type of motion.

• The magnitude of radial acceleration at any instant is v2/r where v is the speed and r is the radius of curvature at an instant. In the case of circular motion, r will be the radius; also the direction of radial acceleration is along the radius of curvature.

• The magnitude of the tangential acceleration is equal to the rate of change of speed of the particle w.r.t. time and it is always tangential to the path.

• The tangential and normal accelerations are perpendicular components of the resultant acceleration so their vector sum returns the resultant acceleration.

• In the case of uniform circular motion or UCM, tangential acceleration is always zero as speed doesn’t change. In other words, the resultant acceleration vector in the case of UCM is orthogonal to the instantaneous velocity.

• A body moving with a constant speed never bears any tangential acceleration regardless of the nature of the path.

• For any rectilinear motion (be it uniform/non-uniform) radial acceleration is always zero. It is because the radius of curvature of a straight line is infinite.

• A body moving along a curved trajectory will have some non-zero radial acceleration.

We  all know tha
t according to Newton’s Law Of Motion, any particular body or object has the tendency to change speed when in motion and this depends on the force and the amount of force that is put on the body or object. Acceleration is the measure of the rate of change in the speed and direction of the particular body or object, where the motion can be either linear or circular.

Linear acceleration refers to the type of acceleration that is involved in linear motion and  the type of acceleration that is  involved in circular motion is known as Angular acceleration.

The acceleration that is directed towards the center is known as Radial Acceleration and is measured in Radians per Square Second and the reason for acceleration is the Centripetal force.

When an object or a body moves with a non-uniform speed, it is tangential acceleration.

Features and Characteristics of Radial Acceleration:

1. Angular acceleration can be divided into Radial and Tangential acceleration

2. Radial acceleration shall be defined as an acceleration of an object that is directed towards the center.

3. If Radians per second square which is represented as ωs-2 is how Radial Acceleration is measured.

4. Radial acceleration is also known as Centripetal Acceleration.

5. The component of angular acceleration tangential to the circular path is what  Tangential Acceleration is.

The word rectilinear literally means “straight” in Geometry and the rectilinear propagation of light means that light travels from the source in a straight line. Due to this property, light does not bend due to which we are unable to look around the corner of objects where the light ray falls upon.

There are two notable phenomenons related to the rectilinear propagation of light- reflection and refraction. Reflection can be demonstrated using a mirror and refraction can be explained when a person puts his hand inside a tub of water, the hand appears bent and smaller. Let’s perform an experiment to understand the rectilinear propagation of light better.

Experiment

Take three pieces of cardboard and pierce a hole through each of them at the same point in such a way that when the three of them are kept together, the three holes are in the same straight line. Now, the cardboards have to be set up on a table at a regular distance between them. Then take a candle that is of the same height as the distance between the tabletop and the hole in the cardboard. Light the candle and observe.

When seen from the hole of the last cardboard, the flame of the candle can be seen through the straight line of holes. But if the position of any of the cardboards is slightly disturbed and a hole is moved away from its position then the light from the candle flame will not be able to travel through and it will not be seen from the other end.

This experiment shows that light travels in a straight line. It also shows that light cannot travel through an opaque medium. Let’s take another example to show that light can travel through other transparent mediums such as water.

Take a glass of water and put a part of one of the fingers in the water. It will be seen that the finger is still visible but it is bent and appears to be smaller. This phenomenon is known as the refraction of light that causes light to bend in such a way that it creates an illusion of a raised body or a shorter hand in the water. Another natural instance of this phenomenon is the twinkling of stars in the sky. The light from the stars suffers refraction in the atmosphere due to difference in densities of the different layers of the atmosphere due to which the star appears to be twinkling even when they do not.

To know more about the rectilinear propagation of light and the various other phenomena associated with it, visit ‘s website and get free study material and resources or download the app for easy access where you will get everything you need.

You must have felt your temperature due to fever during times of illness. During the Covid-19 times, everywhere we visited, our temperatures would be checked. As per the Georgia State University, temperature can be defined as a measure which is the average of kinetic energy of a substance. This temperature is measured using certain scales universally. Today we will study two of them – celsius and kelvin. 

Celsius and Kelvin Scales

Kelvin and Celsius are standard units that are used to measure temperature. They have both similarities and differences. One difference is that their starting point varies. IF K is the temperature on the Kelvin Scale and D is the temperature on the Celsius scale, then 

K = D + 273 or D = K – 273. 

The units for Celsius and Kelvin are degree celsius and the temperature in the kelvin scale itself respectively. 

The scales are used to obtain measurements of temperature which are then used in various calculations and calibrations. The facts and parameters for both celsius and kelvin are equally valid. The value of Kelvin especially stands at -273 which is also the negative reciprocal of 0.0366. It is considered as a reliable thermodynamic measuring scale and because it gives almost errorless results, it is widely used by scientists. While celsius is more familiar to common people, kelvin is used in standardized processes and equations. 

With international agreements certifying the celsius scale, we can use it to measure both the absolute zero and triple point apart from the freezing and boiling points of water. 

Difference Between Celsius and Kelvin Scale

Until 1954, Scientists stated that the melting point of ice is 00 C and the boiling point of ice is 100 0C. All these parameters are considered under constant temperature and pressure. It would help if you learnt all these from your schools.

 

Well, the unit’ degrees Celsius’ is available in the standard Celsius scale. The international agreement has certified this scale. You can calculate the temperature of two different points, such as absolute zero and absolute point. This type of scale is used to calculate the temperature difference of specially prepared water.

 

Temperature measuring units like Celsius and Kelvin both are significant. The use of these two units in particular in the scientific world. In these cases, calibration is essential. Also, these two units possess very much similarity along with distinct quality.

 

These two parameters are widely used among multiple sectors for the proper measurement of instruments. 

Insights from History:

Andres Celsius Put forward the unit ‘Celsius” after his name. He invented the unit in 1742. Just explained before, the boiling point and melting point of ice before 1954 was 1000C and 0oC. These parameters were obtained under standard atmospheric pressure.

 

Scientists considered the obtained measurements for the calculation of multiple problems and any other important calculations or mechanical usages.

 

However, the actual value is -273.15 0C, and the triple point lies at 00C. Many scientists consider the definition for Celsius scale still worthy. Well, let’s not focus on the definition but focus on facts and parameters.

 

Like Celsius, Kelvin was denoted as the degree unit to honor Sir William Thomson the Baron of Kelvin. He was the first person who proposed that there should be another system just like the same size as the unit (Celsius). 

 

Some scientists are predicting the reason behind why it can easily be combined with any type of mathematical equation. Also, the definition has the statement that it is having variables of temperature. 

Calculation of Celsius and Kelvin

The Celsius scale slightly differs from the Kelvin but not that much. First, you may change it to Kelvin. Then the value is ready to apply.  

 

It is quite easy if you consider the conversion as usual. If you want to convert it to Kelvin, add 273.15 to your Celsius value, such as K = °C + 273.15.

 

But in the case of calculation of Celsius, try the subtraction method.

 

Subtract the same constant from Kelvin value to convert it to Celsius such as °C = K − 273.15.

Consider fluid A and another fluid B. One is hair oil and another is milk, each of these is filled in one container. Now, let’s compare the two by pouring them into another container by switching on the timer.

 

Here, we would notice that the milk takes less time as compared to the hair oil, do you know why? It’s because the hair oil is more viscous or it is denser than the milk. So, why do we consider these terms as different when both of these carry the same meaning? 

 

 

Difference Between Viscosity and Density

In the above example, we took two fluids viz: hair oil and milk. Now, let’s understand how viscosity is related to density. 

 

Now, let’s take a look at another example. Consider fluid A as honey and another as water. At the microscopic level, honey has tightly bound particles, whereas water has particles that are far apart. So, when we differentiate in terms of the distance each particle bears from another particle in a fluid, then it is density.

 

Now, let’s understand an example of a pickle. Let’s suppose that you have a big jar of pickles and want to transfer some pieces of it into the small jar, you would notice that the layers of oil come along with each piece and it takes a bit of time to reach another jar. You might have wondered why this happened? Ummm, quite yes!

 

Well, it is because there is friction between the two layers and this friction hampers the fast flow of fluid, i.e., oil and the pickle pieces. So, the friction caused is called viscosity. 

 

Not only the liquid does, but air also has a viscosity that varies with temperature. I believe that through this example, you were able to understand what viscosity density is.

 

The key differences between viscosity and density are given in the following table.

 

Measuring Dynamic Viscosity

A rotational viscometer is one of the more popular types of instruments, and it is used to measure dynamic viscosity. In a liquid sample, this instrument rotates a probe. Viscosity is evaluated by measuring the force or torque needed to rotate the probe.

 

The rotational viscometer is especially useful in measuring non-Newtonian liquids. When non-Newtonian liquids are exposed to different conditions, they change viscosity. For instance, some non-Newtonian liquids increase in viscosity with an increase in applied force, whereas other non-Newtonian liquids show a decrease in viscosity with an increase in applied force.

 

As the probe moves in the liquid, the rotational viscometer adjusts its turning speed. The viscometer determines the variation in the viscosity of the sample as the speed, sometimes referred to as shear rate. The unit of measure for the dynamic viscosity is denoted as Centipoise (cP).

 

Measuring Kinematic Viscosity

There are few methods to find the kinematic viscosity of a fluid. The most common method to measure the kinetic viscosity is by determining the time it takes a fluid to flow through a capillary tube. The time is converted directly to kinematic viscosity with the help of a calibration constant provided for the specific tube. The unit of measure of kinematic viscosity is Centistokes (cSt).

 

A primary difference between the dynamic and kinematic viscosity measurements is density. With density, the conversion between a kinematic and a dynamic viscosity can be carried out. The formula for the conversion from kinematic to dynamic and from dynamic to kinematic is:

 

 

For a given sample, dynamic viscosity will always be the higher number with a density greater than one.

 

Relation Between Viscosity and Density

We don’t find the direct viscosity and density relation; however, both of these are affected by temperature.

 

As we can see, honey during winters has high density because it solidifies and in a solid-state, the interatomic particles are attached. When the same is kept under the sun or when the jar of honey is kept under the vessel containing hot water, the honey melts. So, what we notice is, the interatomic particles make some distance under the effect of rising temperature and also the friction between the layers of honey while pouring it into another bowl reduces.

 

 

The above scenario clearly explains the two following parameters:

 

Viscosity and Density Equation

A subject like Physics does not rely on theoretical knowledge only; it also focuses on mathematical equations, so let’s discuss the density viscosity equation. According to the equation of kinematic viscosity, the equation says the following:

 

[v = frac{mu}{p}]

 

Here,

 

v = Kinematic viscosity.

 

A kinematic viscosity represents the dynamic viscosity of a fluid per unit density. The unit of the kinematic viscosity is m²/s. 

 

Dynamic viscosity = It is a force required to overcome the internal friction of any fluid. The unit employed for measuring the dynamic viscosity of a fluid is Pa.s (where ‘Pa’ stands for Pascal and ‘s’ stands for seconds).

 

μ = absolute viscosity. Absolute viscosity is a parameter for measuring the internal resistance in the fluid. 

 

= Density of the liquid or fluid or air. In MKS (Meter-kilogram-second), the unit of the density is kgm⁻³. In CGS or Centi-gram-second, the unit of the density is gcm⁻³.

 

Units of Kinematic Viscosity

The kinematic viscosity carries the three types of measuring units, let’s discuss these in a tabular form:

 

Units of Absolute Viscosity

An absolute viscosity carries the three types
of measuring units, let’s discuss these in a tabular form:  

 

Dimensional Formula of Viscosity

We know that the formula for the viscosity is:

 

[mu = frac{F}{A}frac{y}{U}]

[= frac{FL}{Lsurd(frac{L}{T})}]

= [frac{FT}{L^{2}}]

 

We write this expression to write the fundamental unit of viscosity. So, rewriting the equation in the following way: 

 

As we know that the dimensional formula for the force, i.e., F is MLT⁻² , so putting this value in equation (1), we get:

 

[=frac{MLT^{-2}T}{L^{2}}]

 

After canceling the common terms, we get the dimensional formula for the viscosity as:

 

μ = MLT⁻¹

 

Dimension of coefficient of viscosity

The coefficient of viscosity is η

 

Dimension of viscous force is [ M L T -2]

 

Dimension of length is [ L ]

 

Dimension of velocity is [ L T -1] 

 

Dimension of the area is L2

 

η = [ M L T -2 L L T -1 L2]

 

η = [ M L-1 T -1 ]

 

What is the density and viscosity of water?

Pure water has its highest density at 4.0°C = 1000 kg/m3.

 

Water – Density Viscosity Specific Weight.

 

How is viscosity determined?

Viscosity is defined as the measure of the resistance of a substance to a motion under an applied force. The viscosity is generally expressed in centipoise (cP), which is the equivalent of 1 mPa s (millipascal second). Shear stress is the force per unit area required to move one layer of fluid in relation to another.

 

How is density determined?

Density is a mass of a unit volume of a material substance. The formula for calculating density is,

 

[d = frac{M}{V}],

 

Where d = density, 

M = mass, 

V = volume.

 

Conclusion

Newtonian liquids have an inherent viscosity that does not change with the change in the force applied to the liquid. This inherent viscosity can be accurately determined with a capillary-type apparatus, using gravity to move the fluid. Alternatively, non-Newtonian fluids exhibit wide variations in viscosity based on the applied force. These require testing with rotational viscometers in order to measure changes over time and over a range of applied forces.

Relative velocity is the velocity of an object in relation to another object. It is a measure of how fast two objects are moving with respect to each other. Relative velocity is important in physics because it helps us understand how objects move and interact with one another. Relative velocity can be measured for many different types of interactions. Relative velocity is the velocity of an object in relation to another object. It is a measure of how fast two objects are moving with respect to each other. Relative velocity is important in physics because it helps us understand how objects move and interact with one another.

Relative velocity can be measured for many different types of interactions. Relative velocity is an important concept in physics that allows us to understand how objects move and interact with one another by measuring the velocity of two objects in relation to each other. Relative velocity can be measured for many different types of interactions which makes it a versatile tool for physics. In physics, the relative velocity is the velocity of an object in relation to another object. It is a measure of how fast two objects are moving with respect to each other. Relative velocity is important in physics because it helps us understand how objects move and interact with one another. 

We know that velocity is a function of time. It is the speed of an object with its direction. However, when we are talking about the velocity of one object with respect to another, that means we are discussing the concept of relative velocity. So, what is relative velocity? Relative velocity is the velocity of an object A with respect to another object B. In simple words, it is the rate of change of relative position of object A with respect to object B. 

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Now, let’s analyze relative velocity.

Relative Velocity in Two Dimensions

Consider two objects, P and Q, traveling with uniform velocities, v1 and v2, along the parallel lines in the same direction. At the time they started, time ‘t’ was zero, and their displacements from the origin were x01 and x02, respectively. 

Now, when the time becomes ‘t’ and their displacements become x1 and x2 with respect to the origin with the position axis, then the equation for object P becomes:

x1 = x01 + v1t…..(1)

Similarly, for object Q, the equation becomes:

x2 = x02 + v2t…..(2)

Subtracting equation (1) from (2), we get:

(x2 – x1) = (x02 – x01) + (v2 – v1)t….(3)

Since x01 and x02 are the initial displacements of the object Q with respect to the object P at time ‘t = 0’, so the equation is:

x0 = x02 – x01….(4)

Now, substituting the value of eq (4) in (3), we get the new equation:

(x2 – x1) = x0 + (v2 – v1)t….(5)

One more thing to note here: (x2 – x1) is the relative displacement of the object Q with respect to the object P at the time ‘t’, so we rewrite equation (5) as:

x = x0 + (VQ – VP)t….(6)

Rearranging equation (6) as:

$frac{x- x_0}{t} = (v_2 – v_1)cdots (7)$

We know that the change in the displacement per unit time is velocity, and the same thing can be observed in equation (7), where LHS equals the RHS. 

Relative Velocity for Objects Moving in the Same Direction

Now, the equation for the relative velocity of the object Q with respect to the object P from equation (7) is:

vQP = v2 – v1 ….(8) 

Relative Velocity for Objects Moving in the Opposite Direction

Now, the equation for the relative velocity of the object Q with respect to the object P from equation (7) is:

vQP = v2 + v1 ….(9) 

Dimension of Relative Velocity

The dimension of relative velocity is the same as that of the velocity, and it is given by:

M⁰L¹T⁻¹

Now, let’s discuss a few questions on relative velocity.

Relative Velocity Problems

Question 1: What Would Happen if Both the Objects Travel with the Same Velocity?

Ans: If both the objects have the same velocity, then,

vQP = v2 – v1.

If v2 = v1, then x – x0 = 0, or x = x0, which means these two objects will remain at the constant distance apart, i.e., their relative distance, and therefore, the position-time graph for the same will be parallel lines. The graph for this condition is drawn below.

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Question 2: Consider the Following Two Cases for Relative Velocities and Express them Mathematically.

1. If v2 < v1

2. If v2 > v1

Ans: In the first case, when v1 > v2, the difference between the velocities will be negative; also, the difference x – x0 will be negative. It means that the separation between the two objects traveling with respect to each other goes on decreasing by the amount v1 – v2 after each time interval. 

In the second case, when v1 < v2 the difference between the velocities will be positive; also, the difference x – x0 will be positive. It means that the separation between the two objects traveling with respect to each other goes on increasing by the amount v1 – v2 after each time interval. 

The graph for both cases is as follows.

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Relative Velocity Examples

Now, let’s have a look at a few examples of relative velocity and relative motion.

< span>Example 1:

Consider a boy running with a relative velocity of vrel→on a train running with a velocity of vt→relative to the ground, so the speed of the boy relative to the ground will be:

v→= vrel→+ vt→

Example 2:

Consider a woman running on the race track in the direction of her competitors (running with a velocity of VC→) with a velocity of vrel→, then the equation for the relative velocity becomes:

v→= vrel→– vc→

 Now, if she moves in the opposite direction, then the equation will be:

v→= vrel→+ vc→

 

Example 3:

If a satellite is moving in the equatorial plane with a velocity of sand at any point on the earth’s surface with a velocity of relative to the center of the earth, then the relative velocity of a satellite with respect to the surface of the earth will be:

vse→= vs→– ve→

 If this satellite moves from the west to east, i.e., in the direction of the rotation of the earth on its axis, then the equation becomes:

vse = vs – ve