2.1.2 Kinematic Graphs

The variation with time of a, v and s for a rectilinear motion are often plotted on graphs. The resulting graphs are called the a-t, v-t and s-t graphs respectively.

Remember that v is the rate of change of displacement.

$\displaystyle v=\frac{{ds}}{{dt}}$

anda is the rate of change of velocity.

$\displaystyle a=\frac{{dv}}{{dt}}$

If we reverse the differentiation by integrating in the opposite direction, we can see that Δv is the time integral of a,

$\displaystyle \int_{{{{t}_{1}}}}^{{{{t}_{2}}}}{{a\text{ }}}dt=\int_{{{{t}_{1}}}}^{{{{t}_{2}}}}{{\frac{{dv}}{{dt}}\text{ }}}dt=\int_{{{{v}_{1}}}}^{{{{v}_{2}}}}{{\text{1 }}}dv=\left[ v \right]_{{{{v}_{1}}}}^{{{{v}_{2}}}}={{v}_{2}}-{{v}_{1}}$

and Δs is the time integral of v.

$\displaystyle \int_{{{{t}_{1}}}}^{{{{t}_{2}}}}{{v\text{ }}}dt=\int_{{{{t}_{1}}}}^{{{{t}_{2}}}}{{\frac{{ds}}{{dt}}\text{ }}}dt=\int_{{{{s}_{1}}}}^{{{{s}_{2}}}}{{\text{1 }}}ds=\left[ s \right]_{{{{s}_{1}}}}^{{{{s}_{2}}}}={{s}_{2}}-{{s}_{1}}$

Never mind the calculus if it intimidates you. The A-level only tests these concepts graphically. All you need to know is that

s-t graph

The gradient of a s-t graph (at a particular instant) represents the velocity (at that instant).

v-t graph

The gradient of a v-t graph (at a particular instant) represents the acceleration (at that instant).
The area under the v-t graph (between two instants in time) represents the change in displacement (during that time interval).

a-t graph

The area under the a-t graph (between two instants in time) represents the change in velocity (during that time interval).

Example

Given the v-t graph for the motion of a ball over 5 seconds, derive the corresponding s-t and a-t graphs. (Assume “rightward is positive” sign convention.)