Graphical Representation of Displacement with Respect to Time
The displacement of a moving body at different instants of time can be represented using a graph. Such a graph is drawn by taking displacement on the Y-axis and time on the X-axis. This curve is called the s–t curve. We consider two important cases:
1. When the body moves with uniform velocity
When a body moves with constant velocity, it covers equal distances in equal intervals of time. Plotting displacement on the Y-axis and time on the X-axis produces a straight-line s–t graph, as shown in Fig. 2.1 (a).
The equation of motion for uniform velocity is:
s = u × t
Thus, velocity at any instant is:
Velocity at t₁ = s₁ / t₁
Velocity at t₂ = s₂ / t₂
Since velocity remains constant:
s₂ – s₁ / t₂ – t₁ = tan θ
Here, tan θ is the slope of the s–t graph. Therefore, the slope of the displacement-time curve gives the velocity of the body.
2. When the body moves with variable velocity
If a body moves with variable velocity, the distances covered in equal intervals of time are not the same. Therefore, the s–t graph becomes a curve, as shown in Fig. 2.1 (b).
Consider a point P on the s–t curve. Let the body move from P to a nearby point Q in a small time Δt, giving a small displacement Δs. Join P and Q to form a chord PQ making an angle θ with the horizontal.
The average velocity during interval PQ is:
tan θ = Δs / Δt
As Q approaches P (i.e., Δt → 0), the chord PQ becomes the tangent at point P. Thus, the velocity at point P is:
v = tan θ = ds/dt
Hence, the slope of the tangent to the s–t curve at any instant gives the instantaneous velocity of the body.
Graphical Representation of Velocity with Respect to Time
Velocity–time graphs (v–t curves) help us understand how the velocity of a moving body changes with time. We consider two important cases:
1. When the body moves with uniform velocity
If a body moves with constant velocity (zero acceleration), the v–t graph becomes a horizontal straight line, as represented by line AB in Fig. 2.2 (a).
Distance covered in time t is equal to the area under the v–t curve.
Distance moved in 1 second = Area under line AB
Since the graph is a straight line:
Distance = Area of rectangle OABC
Hence, the distance covered in any time interval is obtained directly from the area under the v–t graph.
2. When the body moves with variable velocity
When the velocity of a body changes with time (i.e., when the body has uniform acceleration), the v–t graph becomes an inclined straight line, as shown in Fig. 2.2 (b).
The slope of the v–t graph gives the acceleration of the body:
tan θ = (change in velocity) / time = (v – u) / t = a
Thus, from the graph:
v = u + at
Since the distance covered is equal to the area under the v–t curve,
Distance (s) = Area OABD = Area OACD + Area ABC
So,
s = ut + ½ at²
This confirms the standard equation of motion under constant acceleration.
Graphical Representation of Acceleration with Respect to Time
An acceleration–time graph (a–t curve) shows how the acceleration of a moving body varies with time. The area under this curve gives the change in velocity. We consider two cases:
1. When the body moves with uniform acceleration
If a body has constant acceleration, the a–t graph becomes a horizontal straight line, as shown in Fig. 2.3(a). Since the change in velocity is obtained by multiplying acceleration and time, the area under the a–t graph (OABC) represents the total change in velocity.
2. When the body moves with variable acceleration
If acceleration changes with time, the a–t curve may take any shape depending on how acceleration varies at each instant, as shown in Fig. 2.3(b).
At any instant t, if the acceleration of the body is a, then:
a = dv/dt or dv = a dt
Integration of a–t curve
Integrating both sides:
∫v₁v₂ dv = ∫t₁t₂ a dt
or,
v₂ − v₁ = ∫t₁t₂ a dt
Here, v₁ and v₂ are the velocities of the body at the times t₁ and t₂.
The integral on the right-hand side represents the area under the a–t curve between the time intervals t₁ and t₂. Thus, the area under the acceleration curve between any two vertical lines gives the change in velocity.
If the initial velocity is u at time t = 0 and the final velocity is v at time t, then:
This area corresponds to the region OABC under the a–t diagram.