Specialist2025 NHT Exam 2 Section B · Q4CAS active10 marks

Duck and drone: vector motion

A duck swims on a lake. The duck’s position vector, relative to an origin, $O$ , at time $t$ seconds, is given by

$\tvec{r}_{\text{D}}(t) = \left(2 - 5\cos(t)\right)\tvec{i} + \left(1 - 6\sin(t)\right)\tvec{j}, \text{ where } 0 \le t \le 10.$

Find the Cartesian equation of the path of the duck.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

How far does the duck travel along the path in the first 5 seconds?

Give your answer in metres correct to one decimal place.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

The duck is to be monitored by a drone. The drone is dispatched at $t = 0$ and its velocity vector is given by

$\dot{\tvec{r}}_{\text{R}}(t) = (at + 20)\tvec{i} + (bt + 30)\tvec{j} + \left(ct^2 + 2t\right)\tvec{k}$ , where $t \ge 0$ .

Components are measured in metres per second and $a$ , $b$ and $c$ are real constants.

The initial position of the drone is $20\tvec{i} + 30\tvec{j}$ relative to the origin $O$ used for the duck’s position vector.

Use integration to show that the position vector of the drone is

$\tvec{r}_{\text{R}}(t) = \left(\frac{a}{2}t^2 + 20t + 20\right)\tvec{i} + \left(\frac{b}{2}t^2 + 30t + 30\right)\tvec{j} + \left(\frac{c}{3}t^3 + t^2\right)\tvec{k}.$

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Find the values of $a$ , $b$ and $c$ if the drone reaches the duck when $t = 10$ .

Give your answers correct to one decimal place.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

At what time is the duck closest to the initial position of the drone?

Give your answer in seconds correct to one decimal place.

3 marks

Hints, answer checking and worked solutions come with a Season Pass.

Specialist2025 NHT Exam 2 Section B · Q4CAS active10 marks

Duck and drone: vector motion

A duck swims on a lake. The duck’s position vector, relative to an origin, $O$ , at time $t$ seconds, is given by

$\tvec{r}_{\text{D}}(t) = \left(2 - 5\cos(t)\right)\tvec{i} + \left(1 - 6\sin(t)\right)\tvec{j}, \text{ where } 0 \le t \le 10.$

Find the Cartesian equation of the path of the duck.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

How far does the duck travel along the path in the first 5 seconds?

Give your answer in metres correct to one decimal place.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

The duck is to be monitored by a drone. The drone is dispatched at $t = 0$ and its velocity vector is given by

$\dot{\tvec{r}}_{\text{R}}(t) = (at + 20)\tvec{i} + (bt + 30)\tvec{j} + \left(ct^2 + 2t\right)\tvec{k}$ , where $t \ge 0$ .

Components are measured in metres per second and $a$ , $b$ and $c$ are real constants.

The initial position of the drone is $20\tvec{i} + 30\tvec{j}$ relative to the origin $O$ used for the duck’s position vector.

Use integration to show that the position vector of the drone is

$\tvec{r}_{\text{R}}(t) = \left(\frac{a}{2}t^2 + 20t + 20\right)\tvec{i} + \left(\frac{b}{2}t^2 + 30t + 30\right)\tvec{j} + \left(\frac{c}{3}t^3 + t^2\right)\tvec{k}.$

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Find the values of $a$ , $b$ and $c$ if the drone reaches the duck when $t = 10$ .

Give your answers correct to one decimal place.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

At what time is the duck closest to the initial position of the drone?

Give your answer in seconds correct to one decimal place.

3 marks

Hints, answer checking and worked solutions come with a Season Pass.