(a) Distance – Time (x — t) Graph: Velocity
A plot of distance against time is called a distance-time graph. The velocity of a particle at any point can be found from the graph of its position as a function of time. For a uniform motion, the distance-time graph is a straight line as shown in fig 1 (i) whereas for non-uniform motion it is as shown in fig 1 (ii).
The distance-time graphs is used to determine
- Position of a body at any time.
- Distance covered by a body in any interval of time.
- Speed of the body at any instant of time.
- The slope of the x – t graph at any point gives the instantaneous velocity of the body.
Velocity – Time (v – t) Graph: Acceleration
A plot showing the variation of velocity as a function of time is called the velocity-time (v – t) graph. The velocity-time graph for a body moving with constant speed in a straight line is a straight line parallel to the time axis as shown in fig.3(i)
The velocity-time graph for a body moving with a constant acceleration having initial velocity zero is a straight line passing through the origin as shown in fig. 2 (i). However for a body having non-zero initial velocity, the v – t graph look like in fig 2 (i).
The velocity-time graph for a body moving with non-uniform acceleration is as shown in fig 3 (ii)
The v-t graph can be used to determine the acceleration of the body at any instant of time. Referring to fig.3(ii), if we have to find the acceleration at point P, we draw a tangent AB at point P. The slope of tangent BC/AC gives the acceleration. Similarly, this graph can be used to find the velocity of the body at any instant of time.
We know, from the definition, of instantaneous velocity,v = dx/dt.
This equation can be inverted to find the displacement in the same interval of time.
The geometrical meaning of the above equation is that it gives the area under the v-t curve in the interval Δt = t2 – t1 (referring to fig.4). The shaded region represents the displacement of the body in the time interval Δt = t2 – t.