Vectors (Kinematics)

00:00 8 marks

A particle P moves such that its velocity vector at time $t$ seconds ($t\ge0$) is given by

$$
\mathbf{v}(t)=\begin{pmatrix}2t-3\\
\sin(t)\end{pmatrix}\quad\text{ms}^{-1}.
$$

The initial position of the particle is

$$
\mathbf{r}(0)=\begin{pmatrix}1\\ -1\end{pmatrix}\ \text{m}.
$$

(a) Find the speed of the particle when $t=\pi$. [3]

(b) Find the acceleration vector of the particle when $t=2$. [2]

2 minutes 3 marks

Find the speed of the particle when $t=\pi$.

Hint Hint

Compute v(π) and then its magnitude.

|2π − 3| ≈ 3.28 ms⁻¹

3 minutes 2 marks

Find the acceleration vector of the particle when $t=2$.

Hint Hint

Differentiate v(t) to get a(t), then substitute t=2.

(2, cos 2) ≈ (2, −0.416) ms⁻²

3 minutes 3 marks

Find the position vector $\mathbf{r}(t)$ of the particle at time $t$.

Hint Hint

Integrate v(t) componentwise, then use the initial condition to determine constants.

$\mathbf{r}(t)=(t^2 – 3t + 1, -\cos t)$