Graphical Representation of Displacement with Respect to Time

The displacement of a moving object in a given time may be given by means of a graph. Such a graph is drawn by plotting the displacement as ordinate and the abscissa as corresponding time. We will discuss the following two cases:

When the object moves with uniform velocity. When object moves with uniform velocity, equal distances are covered in equal time intervals. By plotting the distances on Y-axis and time on X-axis, a displacement time curve is drawn which is a straight line, as shown in Figure. Therefore motion of the body is governed by the equation s = u.t, such that

Velocity at instant 1 = s1 / t
1
Velocity at instant 2 = s2 / t
2

Since the velocity is uniform, therefore

where tan θ is called the slope of s-t curve. In other words, the slope of the s-t curve at any instant
gives the velocity.

When the object moves with variable velocity. When the object moves with variable velocity, unequal distances are covered in equal time intervals or equal distances are covered in unequal intervals of time. Thus the displacement-time graph, for such a case, will be a curve, as shown in Figure.

Consider a point P on the s-t curve and let this point travels to Q by a small distance δs in a small time interval δt. Let the chord joining the points P and Q makes an angle θ with the horizontal axis. The average velocity of the moving point during the interval PQ is given by

tan θ = δs / δt. …(From triangle PQR )

In the limit, when δt approaches to zero, the point Q will tend to approach P and the chord PQ
becomes tangent to the curve at point P. Thus the velocity at point P,

Vp = tan θ = ds /dt

where tan θ is the slope of the tangent at point P. Thus the slope of the tangent at any instant on the s-t curve gives the velocity at that instant.

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