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.