1. A particle has position vector at time $t$ equal to $5t^2\mathbf{i} + (2t-t^2)\mathbf{j}$. Find the exact magnitude of the acceleration of the particle.
$\sqrt{104}$
2. A particle has acceleration at time $t$ equal to $(4t-1)\mathbf{i} + t^2\mathbf{j}$. The particle is instantaneously at rest when $t = 0$. Find the speed of the particle when $t = 3$. Write your answer in the form $k\sqrt{34}$.
$3\sqrt{34}$
3. A particle has position vector $\mathbf{r} = t^2\mathbf{i} + 6t\sqrt{t}\mathbf{j}$. Find the exact speed of the particle at $t = 1$.
$\sqrt{85}$
4. A particle is moving in a single direction and has acceleration $a = 1 - \sin t$ m s-2 at time $t$. The particle starts from rest at the origin. Find an expression for the displacement of the particle at time $t$ seconds.
$\dfrac{1}{2}t^2-t+\sin t$
5. A particle starts from rest and, at time $t$, has acceleration $3t^2\mathbf{i} + (1-2t^3)\mathbf{j}$. Find the velocity of the particle when $t = 2$.
$8\mathbf{i}-6\mathbf{j}$
6. The displacement of a particle at time $t$ seconds is given by $s = 2t\sqrt{t} + \mathrm{e}^{-t}$ m. Find the acceleration of the particle at $t = 2$.
1.20
7. A particle has acceleration equal to $3\sqrt{t}$ m s-2 at time $t$ seconds. Its velocity is 2 m s-1 at $t = 1$. Find the time taken for the particle to travel 25 m.
3.96
8. The position vector of a particle is given by $\mathbf{r} = (t^2 - 2t)\mathbf{i} + 5t\mathbf{j}$. Find the value of $t$ when the particle has a speed of 6.
2.66
9. A particle has acceleration at time $t$ given by $\mathbf{a} = (1-4t)\mathbf{i} + (3-t^2)\mathbf{j}$. At $t = 0$, the velocity of the particle is $24\mathbf{i} + 6\mathbf{j}$. Find the velocity of the particle at $t = 3$.
$9\mathbf{i}+6\mathbf{j}$
10. A particle has acceleration $(5t-3)\mathbf{i} + (8-t)\mathbf{j}$. At $t=0$, the particle has velocity $2\mathbf{i} - 5\mathbf{j}$. Find the speed of the particle when it is moving parallel to the vector $\mathbf{i}-\mathbf{j}$.
1.59
11. A particle has acceleration at time $t$ given by $(4t-3)\mathbf{i} - 6t^2\mathbf{j}$. At time $t = 0$ the particle has velocity $2\mathbf{i} + 3\mathbf{j}$. How far away from its starting location is the particle at time $t = 1$?
2.76
12. A particle has velocity $\mathbf{v} = \sin 2t\mathbf{i} + 2t\mathbf{j}$ at time $t$. At $t=0$, the particle has position vector $2\mathbf{i}$. Find the position vector of the particle at $t = \pi$.
$2\mathbf{i}+\pi^2\mathbf{j}$
13. A particle of mass 2 kg is moving under the action of a force $\mathbf{F}$. The velocity of the particle at time $t$ is $\mathbf{v} = (pt^2-3t)\mathbf{i} + 8t\mathbf{j}$. At $t = 0.5$, $|\mathbf{F}| = 20$. Find the two possible values of $p$.
$-3, 9$
14. At time $t=0$, $P$ has position vector $\mathbf{i} + 2\mathbf{j}$ and $Q$ has position vector $11\mathbf{i} + 5\mathbf{j}$. The velocity of $P$ is given by $(4t-3)\mathbf{i} + 4\mathbf{j}$, and the velocity of $Q$ is given by $5\mathbf{i} + k\mathbf{j}$. Given that $P$ and $Q$ collide, find $k$.
3.4
15. A particle has velocity $3\mathbf{i} + 13\mathbf{j}$ m s-1 at time $t = 0$. The acceleration of the particle is $-t\mathbf{i} + 2\mathbf{j}$. Find the value of $t$ when the particle is moving in a north-west direction.
$t=8$
16. The position, $x$, of a particle moving along the $x$ axis at time $t$ seconds is given by $x = (20-t^2)\sqrt{t+1}$. Find the speed of the particle when it passes the origin.
20.9
17. A particle has position vector $\mathbf{r} = (t^2 + 2)\mathbf{i} + (t^3 + kt^2)\mathbf{j}$, where $k$ is a constant. At $t = 3$, the particle has speed $15\sqrt{5}$. Find the greatest magnitude of the acceleration of the particle at $t = 0.5$
17.1
18. A particle has velocity equal to $3 + 8\sin kt$ at time $t$ seconds. Given that the initial acceleration is 4, find the distance between the starting position of the particle and the position of the particle at time $t = \dfrac{\pi}{3}$.
