Graphical Representation of Motion Physics Notes

Graphical Representation of Motion Physics Notes

Graphical Representation of Motion:
1. A graphical representation is a pictorial representation of the relation between two sets of data of which one set is of a dependent variable and other set is of independent variables. Now here dependent variable is shown on Y-axis and independent variable is shown on X-axis.

2. To describe the motion of an object we can use line graphs. In this case line graph shows dependence of one physical quantity, such as distance (position), velocity, acceleration, on the another quantity such as time.

3. Now the slope of graph and area from figure provides other physical quantities. The slope of graph is calculated by tanθ which is the ratio of physical quantities taken on Y-axis to the X-axis. While by the area of figure, the product of quantities taken from the X axis and Y-axis which are dependent upon the figure and its shape.

Now here we showed the three types of graphical representation of motion.
Dependent Quantities

Independent Quantity

1. Position-time graph
2. Velocity-time graph
3. Acceleration-time.

Position-Time Graph:

1. The change in the position of an object with time can be represented on the position-time graph.
2. In this graph, time is taken on the X-axis and position is taken along the Y-axis.
3. The Position-time (P – T) graph of a moving body can be used to calculate the speed of the body as they specifically represent velocity.
4. The slope of the tangent at any point of position¬time graph denotes the instantaneous velocity of the object at that point or tan θ = v

(a) When the object is at rest
Its position will not change with time. Let the object be stationary at position x(t) = x0 from the origin. Then the position (x) time (t) graph for the stationary object is a straight line AB parallel to the time-axis.

Object at rest
For example: (i) A train standing at the railway track or line.
(ii) A bus standing on the road side.
Here θ = 0°
then tan θ° = v
so v = 0 {v ∴ velocity}

(b) When the object moves with a uniform motion:
Uniform motion: Uniform motion is defined as equal displacement occurring during successive equal time periods (some times called constant velocity).

The graph of position time for uniform motion in a straight line that the slope or velocity is constant.

(a) When the object is moving with a positive velocity then the slope of the position-time graph will be tan 0 and is positive then v is constant but positive and acceleration (a) = 0.

Slope of tan θ = \(\frac{y \text { axis }}{x \text { axis }}=\frac{x}{t}\) = velocity
velocity = v = constant
but a = \(\frac{d v}{d t}=\frac{d}{d t}\) (const)
a = 0

(b) When the object is moving with a negative velocity then the slope of position time graph will be negative. Hence θ = constant but θ > 90°, negative tan θ then v is constant but acceleration (a) = 0.

(c) When the object moves with a variable velocity or Non-Uniform Motion: Figure (a) depicts the non-uniform motion in which the graph is not a straight line in the position-time graph. When we calculate the slope at two points P and Q in the graph we find out that the slope at Q is more than the slope at P [tanθ₂ > tanθ₁]. Hence, the velocity at Q is more than the velocity at P.

In this way, the velocity of the particle is increasing with time. Hence, the acceleration of the particle will be positive. Similarly, figure (b) shows the non-uniform motion in the position-time graph. But here when the slopes are calculated at points P and Q; the slope at Q is less than the slope at P; which shows the decreasing velocity and depicts negative acceleration.

Non Uniform Motion

Velocity-Time Graph:
With the help of a velocity-time graph we get the knowledge about the acceleration and displacement of the particle.
1. Uniform velocity motion: Consider that an object is moving with uniform velocity v. Since the object is in uniform motion, the magnitude of it’s velocity at t = 0, t = 1s, t = 2s …. will always be v therefore graph between time and the velocity of the object will be as shown in the figure

Displacement of the object in a given time interval
Consider the two points A and B on the v – t graph corresponding to instants t₁ and t₂ respectively.
Then
Area ABB’A’= v(t₂ – t₁) …(1)

As given by equation (1), the displacement of the object in the time interval between t₁ and t₂ is
x₂ – x₁ = v(t₂ – t₁) ….(2)

It means the displacement of an object in the time interval (t₂ – t₁) is numerically equal to the area under velocity-time graph between the instant t₁ and t₂.

Note: It may be pointed out that this geometrical method of finding the displacement of an object holds good even in the case, when the object is moving with negative velocity. In such a case, the area below the velocity-time graph is taken as negative and corresponding to this, the displacement will also be negative.

2. Constant/uniform acceleration motion:
The slope of a velocity slope of the graph – tanθ – y/x time graph represents the acceleration of the object. So, the value of the slope at a particular time represents the acceleration of the object at that instant.
The slope of a velocity-time graph will be given by the following formula.
Slope of graph = tanθ = y/x time = \(\frac{v_{2}-v_{1}}{t_{2}-t_{1}}=\frac{\Delta v}{\Delta t}\)

Since \(\frac{\Delta v}{\Delta t}\) is the definition of acceleration, the At the slope of a velocity-time graph must be equal to the acceleration of the object.
Slope = acceleration

This means that when the slope is steep, the object will be changing velocity rapidly. When the slope is shallow, the object will not be changing its velocity as rapidly. This also means that acceleration will be negative if the slope is negative. Directed downwards the acceleration will be negative and if the slope is positively directed upwards the acceleration will be positive.

Uniform Acceleration Motion

3. Non-Uniform Acceleration Motion: In the figure, the velocity-time graph shows not a straight line motion. When we calculate the slope at two points P and Q in the graph we find out that the slope at Q is more than the slope at P which means that the acceleration at Q is more than the acceleration at P. In this way this graph shows non-uniform motion whose acceleration is increasing. Similarly, figure (b) shows the non-uniform motion in the

Non Uniform Acceleration Motion

velocity-time graph. But when the slopes are calculated at points P and Q, the slope at Q is less than the slope at P; which shows that the acceleration is decreasing.

Circular Sine Velocity-Time Graph:
In the figure, the velocity-time graph is a line graph. This type of motion generally depicts simple harmonic motion. In this type of motion velocity changes with time in the form of a sine function. By this graph we can easily understand positive and negative displacement. In the given graph from O to T and 2T to 3T time interval velocity is positive. Whereas, T to 2T time interval velocity is negative and the area (S₂) in the graph for this interval is also negative. Therefore, displacement from 0 to 3T time interval is:

Circular Sine Velocity Time Graph
S = S₁ – S₂ + S₃
Whereas distance covered in the time interval in 0 to 3 T is
S = S₁ + S₂ + S₃

Acceleration-Time Graph:
Similar to position-time graph and velocity-time graph acceleration-time graph also gives various information regarding the motion of a particle. In figure (a) acceleration is constant with time which depicts uniform accelerated motion. In figure (b) acceleration is changing with time which shows non-uniform accelerated motion. The change in the acceleration is a straight line or not in straight-line motion both shows non-uniform accelerated motion.

(a) Uniform Acceleration Motion

The area between the two-time intervals in the acceleration-time graph tells us about the change in velocity. As in the figure the shaded area between the time interval t₁ and t₂ tells about the change in velocity;
Δv = Area of shaded part.

(b) Non-Uniform Acceleration Motion

Physics Notes