Graphical Representation of Motion

Concept of motion and it's graphical representation has always been a jargon and a pain to students of class 9th,10th,11th and 12th, be it RBSE or CBSE. Even for cracking IIT and NEET this concept has to be done well. We at Alchemy Learning always insist on getting the concept right.

Distance – Time Graph:

When an object is moving with uniform velocity, the slope of graph is always a straight line. In other words, slope of straight line of a distance-time graph shows that object is moving with uniform speed.

In the above graph, straight slope line shows that object is moving with uniform velocity. Slope OA shows the speed of the object.

Calculation of Speed using distance-time graph:

To calculate the SPEED, let take two points A and B on the slope OB.

Draw a line parallel to y-axis and another parallel to x-axis from B.

Again draw a line parallel to y-axis and another parallel to x-axis from point A.

Let, line parallel to x-axis from point B cut at a point, S₂ at y-axis.

Line parallel to x-axis from point A cut at point, S₁ at y-axis.

   Let, line parallel to y-axis from point B cut at t₂ at x-axis.

Line parallel to y-axis from point A cut at t₁ at x-axis.

Now, BC= Distance = S₂ – S₁

 and  AC = time = t₂ – t₁

We know that slope of the graph is given by the ratio of change in y-axis and change in X-axis.

Or, 

  where, v= speed,

S₂ – S₁ = interval of distance and t₂ – t₁= time interval

Distance – Time Graph of a body moving with Accelerated motion:

When graph of distance Vs time is plotted for an object moving with accelerated motion, i.e. with increasing non-uniform speed, the slope of graph will not be a straight line. The rising trend of slope shows the increasing trend of velocity.

Velocity-Time Graph:

Velocity-time graph of an object moving with uniform velocity:

The slope of a Velocity–time graph of an object moving in rectilinear motion with uniform velocity is straight line and parallel to x-axis when velocity is taken along y-axis and time is taken on x- axis

Calculation of distance using velocity-time graph:

Let two points A and B on the slope of graph.

Draw two lines parallel to y-axis AC from point A, and BD from point B.

Let point D at the x-axis (time axis) is t2 and point C is t1.

Let AB meet at ‘v’ at y-axis, i.e. object is moving with a velocity, v.

Thus, distance or displacement by the object is equal to the area of the rectangle (shaded) ABCD.

Thus, Area of ABCD=BD×DC

⇒ s=v(t₂ – t₁)

Since given object is moving with constant velocity along a straight line, thus displacement will be equal to distance covered.

Therefore, Distance or Displacement = velocity X time interval.

Velocity – Time Graph of an object moving with uniform acceleration

When velocity – time graph is plotted for an object moving with uniform acceleration, the slope of the graph is a straight line.

The pattern of slope of the graph shows that object is moving with uniform acceleration.

Calculation of Displacement and Distance covered by the moving object using velocity time graph:

Let take two points, A and B at the slope of the graph.

Draw a line from B to BD and another from point A to AE parallel to y-axis.

Let AD meets at t2 and AEAE at t1 on the time axis.

Thus, Distance covered by the object in the given time interval (t2–t1) is given by the area of ABCDE.

Therefore, Distance (s) = Area of ΔABC + Area of ACDE

⇒S=1/2 (BC X AC)+(AE+ED)

Displacement of the object during the given time interval (t2−t1) = Area of ACDE

Thus, Displacement =AE×ED

Velocity time graph of an object moving with uniform decreasing velocity:

The slope of the velocity time graph of an object moving with uniform decreasing velocity with uniform acceleration is a downwards straight line. The straight downward slope shows the decreasing velocity with uniform acceleration, i.e. retardation.

Velocity time graph of an object moving with non-uniform velocity:

Equations of Uniformly Accelerated Motion:

There are three equations of motion for bodies travelling with uniform acceleration. These are explained below:

1.First Equation of Motion:(Velocity-Time Relation)

             v = u +at

Where, v = Final velocity of body

             u = Initial velocity of body

            a = Acceleration

And     t = Time

Using formula for acceleration:

2. Second Equation of Motion:(Position-Time Relation)

3.  Third Equation of Motion:

Equations of Motion by Graphical Methods:

Consider the following velocity time graph:

1. First Equation of Motion:

            v = u +at

2. Second Equation of Motion:

Let s be the displacement covered.

As we know that distance covered by an object is given as area enclosed by the graph,

i.e., Displacement, s = Area of quadrilateral OABC

3. Third Equation of Motion:

            v² – u² = 2as

Here, Displacement, s = Area of trapezium OABC

 s = Area of rectangle ADCO + Area of triangle ABD

NOTE:

ü  Area under the graph speed-time gives distance travelled.

ü  Slope of the graph speed-time gives acceleration.

ü  Area under the graph velocity-time gives displacement.

ü  Slope of the graph velocity-time gives acceleration.

ü  Slope of the graph distance-time gives speed.

ü  Slope of the graph displacement-time gives velocity.

ü  Area under the graph acceleration-time gives change in velocity/speed.

 

Motion along a circular path:

Motion of an object along a circular path is called circular motion. Since, on a circular path the direction of the object is changing continuously to keep it on the path, the motion of the object is called accelerated motion.

Uniform circular motion:

• When an object moves in a circular path at a constant speed then motion of the object is called uniform circular motion.
• In our everyday life, we came across many examples of circular motion for example cars going round the circular track and many more. Also earth and other planets revolve around the sun in a roughly circular orbits.
• If the speed of motion is constant for a particle moving in a circular motion still the particles accelerates because of constantly changing direction of the velocity.
• If an object moves in a circular path with uniform speed, its motion is called uniform circular motion.
• Here in circular motion, we use angular velocity in place of velocity we used while studying linear motion.
• Force which is needed to make body travel in a circular path is called centripetal force.

We know that the circumference of a circle of radius r is given by 2πr. If the body takes t seconds to go once around the circular path of radius r, the velocity v is given by v=2πr/t

• One thing we must keep in mind is that uniform linear motion is not accelerated but uniform circular motion is accelerated motion.
• Motion of earth around the sun, motion of moon around the earth, motion of a top, motion of blades of an electric fan, etc. are the examples of circular motion.

Relative Motion:

The term ‘with respect to’ we used earlier has a significance. Let’s take an example, you are sitting on your berth inside a compartment of a moving train. Your distances from the walls, ceiling, floor, windows of compartment etc. are not changing as time passes. This means that with respect to the compartment, your position is not changing. So you are at rest with respect to the compartment. But your distances from a pole on the platform is changing as time passes. So, you are moving with respect to the platform. This means that an object can be at rest with respect to one and in motion with respect to another at the same time. So, the motion is not absolute, rather it is relative. It is simply how do we perceive the change in position of objects as time passes.

Now, what do you observe when the train starts moving? You think that platform is going away. This is true with respect to you and the compartment. The distance of the platform from you and the compartment is changing as time passes. So, it looks like the platform is moving as seen from the compartment. Also, the compartment is moving with respect to the platform.

Suppose two cars A and B are moving on a highway side by side in the same direction. Both started together and are moving equally fast. Then the distance between A and B will remain constant with time and they will perceive the other one as stationary.

Suppose two cars A and B are moving in opposite direction then the velocity of A as seen by B will be addition of their velocities (v_A+v_B) and velocity of B as seen by A is v_A+v_B.

Suppose the cars are moving in same direction but with different velocities v_A and v_B respectively. Then the velocity of A as seen by B is v_A-v_B and velocity of B as seen by A is v_B-v_A.