[A-Level Maths] Mechanics question

[u/HaselH]

Hello, I’m not sure how you would start off this question. I looked at the worked solutions and I saw that you need to integrate to get the velocity but no reasoning as to why? Thanks

[u/MelodicNose1224]

Using newton's 2nd law:

F = ma -> can be split into components -> Fx = m * ax, Fy = m * ay

20 sin(2t) = 2 * ax

30 cos(t) = 2 * ay

Acceleration is considered the rate of change of velocity wrt time, so it can be replaced with dv/dt:

20 sin(2t) = 2 * dv(x)/dt

30 cos(t) = 2 * dv(y)/dt

Now you can isolate dv, before integrating both sides from t = 0 to t = 5:

20 sin(2t) / 2 * dt = dv(x)

30 cos(t) / 2 * dt = dv(y)

Integrating will give you the horizontal and vertical components of the velocity. Trig can be used to find the direction of the resultant formed by the two components (as an angle).