Quick questions on Integration in kinematics: N(A)-Level Additional Mathematics

Since acceleration is the rate of change of velocity, integrating acceleration recovers velocity:

Likewise, integrating velocity recovers displacement:

Each integration brings a constant $+ c$. To find it, substitute a known value, typically given at $t = 0$:

To find the distance travelled between two times, you can use a definite integral of the velocity between those limits (taking magnitudes if the velocity changes sign). This connects directly to the area-under-a-curve idea, since distance is the area under the velocity-time graph.

The constants must be found from the given values; leaving $c$ in the answer loses marks.

Given $v = 2t$, and $s = 0$ when $t = 0$, find $s$. [2 marks]

Given $a = 4$, and $v = 3$ when $t = 0$, find $v$. [2 marks]

Given $v = 3t^2$, find the distance travelled from $t = 0$ to $t = 2$. [3 marks]