Specialist2025 Exam 2 Section B · Q4CAS active10 marks

Motion along a parametric vector path

The path of a moving particle with position vector

$\tvec{r}(t) = \left(5\cos(t) - 4\cos\left(\frac{5t}{2}\right)\right)\tvec{i} + \left(5\sin(t) - 4\sin\left(\frac{5t}{2}\right)\right)\tvec{j}$

Write down the coordinates of the particle's starting point.

1 mark

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

On the graph above, draw an arrow from the point $(9, 0)$ to indicate the direction of motion of the particle.

1 mark

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

Find the value of $t$ for which the particle will first return to its starting point.

1 mark

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

Show that the speed of the particle, in m s $^{-1}$ , at time $t$ can be expressed as $\sqrt{125 - 100\cos\left(\frac{3t}{2}\right)}$ .

3 marks

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

What is the maximum speed of the particle in m s $^{-1}$ ?

1 mark

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

On the graph on page 18, trace the path of the particle for $t \in [0, \pi]$ .

1 mark

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

Specialist2025 Exam 2 Section B · Q4CAS active10 marks

The path of a moving particle with position vector

$\tvec{r}(t) = \left(5\cos(t) - 4\cos\left(\frac{5t}{2}\right)\right)\tvec{i} + \left(5\sin(t) - 4\sin\left(\frac{5t}{2}\right)\right)\tvec{j}$

Write down the coordinates of the particle's starting point.

1 mark

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

On the graph above, draw an arrow from the point $(9, 0)$ to indicate the direction of motion of the particle.

1 mark

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

Find the value of $t$ for which the particle will first return to its starting point.

1 mark

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

Show that the speed of the particle, in m s $^{-1}$ , at time $t$ can be expressed as $\sqrt{125 - 100\cos\left(\frac{3t}{2}\right)}$ .

3 marks

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

What is the maximum speed of the particle in m s $^{-1}$ ?

1 mark

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

On the graph on page 18, trace the path of the particle for $t \in [0, \pi]$ .

1 mark

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