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Question 1

HLPaper 1

The position of a particle is given by

\vec{r}(t)=\left[\begin{array}{c} 2 t^{2}-3 t \\ t^{3} \end{array}\right]

1.

Find the acceleration vector $\vec{a}(t)$ at time $t$ .

[3]

Question 2

HLPaper 1

A particle moves such that its position at time $t$ is given by

\vec{r}(t)=\begin{pmatrix} 4 t \\ 5 t+2 \end{pmatrix}

1.

Find the speed of the particle.

[2]

Question 3

HLPaper 1

Two particles A and B have position vectors:

\vec{r}_{A}(t)=\begin{pmatrix} 2 t+1 \\ 3 \end{pmatrix}, \quad \vec{r}_{B}(t)=\begin{pmatrix} 5-t \\ 1+2 t \end{pmatrix}

1.

Determine whether the particles ever meet.

[6]

Question 4

HLPaper 1

The position vector of a particle, $P$ , relative to a fixed origin $O$ at time $t$ is given by $\overrightarrow{OP} = \begin{pmatrix} \sin(t^2) \\ \cos(t^2) \end{pmatrix}$ .

1.

Find the velocity vector of $P$ .

[2]

2.

Show that the acceleration vector of $P$ is never parallel to the position vector of $P$ .

[5]

Question 5

HLPaper 1

A particle has velocity

\vec{v}(t) = \begin{pmatrix} 6 \cos t \\ -6 \sin t \end{pmatrix}, \quad \vec{r}(0) = \begin{pmatrix} 0 \\ 0 \end{pmatrix}

1.

Find the position vector $\vec{r}(t)$ and describe the path of the particle.

[6]

Question 6

HLPaper 1

A particle P moves with velocity $\boldsymbol{v} = \begin{pmatrix} -15 \\ 2 \\ 4 \end{pmatrix}$ in a magnetic field, $\boldsymbol{B} = \begin{pmatrix} 0 \\ d \\ 1 \end{pmatrix}$, $d \in \mathbb{R}$.

1.

Given that $\boldsymbol{v}$ is perpendicular to $\boldsymbol{B}$, find the value of $d$.

[2]

2.

The force, $\boldsymbol{F}$, produced by P moving in the magnetic field is given by the vector equation $\boldsymbol{F} = a \boldsymbol{v} \times \boldsymbol{B}$, $a \in \mathbb{R}^+$.

Given that $|\boldsymbol{F}| = 14$, find the value of $a$.

[4]

Question 7

HLPaper 2

At an archery tournament, a particular competition sees a ball launched into the air while an archer attempts to hit it with an arrow.

The path of the ball is modelled by the equation $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 0 \end{pmatrix} + t \begin{pmatrix} u_x \\ u_y - 5t \end{pmatrix}$ where $x$ is the horizontal displacement from the archer and $y$ is the vertical displacement from the ground, both measured in metres, and $t$ is the time, in seconds, since the ball was launched.

$u_x$ is the horizontal component of the initial velocity
$u_y$ is the vertical component of the initial velocity.

In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that $u_x = 8$ and $u_y = 10$ .

An archer releases an arrow from the point $(0, 2)$ . The arrow is modelled as travelling in a straight line, in the same plane as the ball, with speed $60 \text{ m s}^{-1}$ and an angle of elevation of $10^\circ$ .

1.

Find the initial speed of the ball.

[2]

2.

Find the angle of elevation of the ball as it is launched.

[2]

3.

Find the maximum height reached by the ball.

[3]

4.

Assuming that the ground is horizontal and the ball is not hit by the arrow, find the $x$ -coordinate of the point where the ball lands.

[3]

5.

For the path of the ball, find an expression for $y$ in terms of $x$ .

[3]

6.

Determine the two positions where the path of the arrow intersects the path of the ball.

[4]

7.

Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.

[4]

Question 8

HLPaper 2

At an archery tournament, a particular competition sees a ball launched into the air while an archer attempts to hit it with an arrow. The path of the ball is modelled by the equation $\binom{x}{y} = \binom{5}{0} + t \binom{u_x}{u_y - 5t}$ where $x$ is the horizontal displacement from the archer and $y$ is the vertical displacement from the ground, both measured in metres, and $t$ is the time, in seconds, since the ball was launched.

$u_x$ is the horizontal component of the initial velocity
$u_y$ is the vertical component of the initial velocity. In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that $u_x = 8$ and $u_y = 10$

1.

Find the initial speed of the ball.

[2]

2.

Find the angle of elevation of the ball as it is launched.

[2]

3.

Find the maximum height reached by the ball.

[3]

4.

Assuming that the ground is horizontal and the ball is not hit by the arrow, find the $x$ coordinate of the point where the ball lands.

[3]

5.

For the path of the ball, find an expression for $y$ in terms of $x$ .

[3]

6.

An archer releases an arrow from the point (0, 2). The arrow is modelled as travelling in a straight line, in the same plane as the ball, with speed 60 m s⁻¹ and an angle of elevation of 10°. Determine the two positions where the path of the arrow intersects the path of the ball.

[4]

7.

Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.

[4]

Question 9

HLPaper 1

A particle starts from the position $\mathbf{r}_0 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$ and moves with a constant velocity $\mathbf{v} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}$ .

1.

Write the vector equation of the particle’s motion.

[2]

2.

Find the position of the particle at $t = 5$ seconds.

[2]

3.

Determine the time when the particle crosses the x-axis (where $y = 0$ ).

[4]

Question 10

HLPaper 1

A particle has position vector

\vec{r}(t) = \begin{pmatrix} 4t \\ t^2 \end{pmatrix}

1.

Show that the particle is moving on a curve and find the angle between the velocity and acceleration vectors at $t = 2$ .

[7]

AHL 3.12—Vector applications to kinematics Questionbank