Quick questions on Kinematics with calculus and projectile motion - WJEC A-Level Maths

When acceleration is not constant, the suvat equations do not apply, so use calculus instead.

Displacement, velocity and acceleration can be written as vectors, for example $\mathbf{r} = (t^2)\mathbf{i} + (3t)\mathbf{j}$. Differentiate each component separately to get the velocity and acceleration vectors. The speed is the magnitude of the velocity vector, $\sqrt{v_x^2 + v_y^2}$. For example, if $\mathbf{v} = (4t)\mathbf{i} + (3)\mathbf{j}$, then at $t = 1$ the velocity is $4\mathbf{i} + 3\mathbf{j}$ and the speed is $\sqrt{16 + 9} = 5\,\text{m s}^{-1}$.

A projectile moves under gravity alone. Resolve the launch velocity into $u_x = u\cos\theta$ and $u_y = u\sin\theta$.

A particle has displacement $s = 2t^2 - t$. Find its velocity at $t = 3$. [2 marks]

A projectile is launched at $30\,\text{m s}^{-1}$ at $60^{\circ}$. Find the vertical component of the initial velocity. [2 marks]

A particle has acceleration $a = 6t$. Given $v = 2$ at $t = 0$, find $v$ as a function of $t$. [3 marks]