This lesson includes 81 additional question variations for subscribers.

# Lesson Worksheet: Linear Motion with Derivatives Mathematics

In this worksheet, we will practice using differentiation to find the instantaneous velocity, speed, and acceleration of a particle.

**Q1: **

A particle starts from a fixed point and moves in a straight line. The distance traveled at time seconds is given by , where .

Find an expression for the speed of the particle at time seconds.

- A m/s
- B m/s
- C m/s
- D m/s
- E m/s

Find the maximum speed of the particle.

- A m/s
- B m/s
- C m/s
- D1 m/s
- E m/s

**Q2: **

A particle of mass 2 kg moves on the positive . At time seconds, the particleβs displacement, , from the origin is given by , where .

Find the velocity of the particle when , giving your answer to 3 decimal places.

The particle is acted upon by a force of variable magnitude, N, which acts in the direction of the positive .

Find the value of when , giving your answer to 3 decimal places.

**Q3: **

A particle
P moves in a straight line. At time
seconds, where
, the acceleration of
P is
m/s^{2} and the velocity
m/s is given by
, where
is a constant. The initial acceleration of
P is
10 m/s^{2}.

Find the value of .

- A3
- B5
- C4
- D
- E

Using the value of found in part 1, work out, in terms of , the values of in the interval for which .

- A and
- B and
- C and
- D and
- E and

Find the maximum velocity and maximum acceleration.

- AMaximum velocity = 6 m/s, maximum acceleration = 10 m/s
^{2} - BMaximum velocity = 2 m/s, maximum acceleration = 10 m/s
^{2} - CMaximum velocity = 10 m/s, maximum acceleration = 6 m/s
^{2} - DMaximum velocity = 6 m/s, maximum acceleration = 0 m/s
^{2} - EMaximum velocity = 10 m/s, maximum acceleration = 2 m/s
^{2}

**Q4: **

At time seconds, a particle has position vector relative to a fixed origin , where , .

Find the velocity of when .

- A m/s
- B m/s
- C m/s
- D m/s
- E m/s

Find the acceleration of when .

- A m/s
^{2} - B m/s
^{2} - C m/s
^{2} - D m/s
^{2} - E m/s
^{2}

**Q5: **

At time seconds, a particle has a position vector relative to a fixed origin , where and .

Find the speed of when .

Find the acceleration of and explain why it is constant.

- A. The acceleration is constant because it is independent of .
- B. The acceleration is constant because it is independent of .
- C. The acceleration is constant because it is independent of .
- D. The acceleration is constant because only one component is dependent on .
- E. The acceleration is constant because only one component is dependent on .

Find the magnitude of the acceleration, giving your answer in exact form.

- A
- B
- C
- D
- E

**Q6: **

A particle moves on the . At time seconds, the velocity of is m/s in the positive -direction, where is given by

Find the acceleration of when .

Find the acceleration of when .

Find the maximum velocity of .

**Q7: **

A particle P of mass 0.25 kg is acted upon by a single force N. Its position relative to a fixed origin at time seconds is m, where , .

Find the speed of P when , giving your answer to 2 decimal places.

Find when .

- A N
- B N
- C N
- D N
- E N

**Q8: **

A particle of mass 500 grams moving on a plane is acted upon by a variable force N. Its velocity at time seconds is given by , .

Find when .

- A N
- B N
- C N
- D N
- E N

**Q9: **

A particle moves in a straight line so that, at time seconds, its displacement, m, from a fixed point is given by

Find the velocity of when .

- A m/s
- B m/s
- C5 m/s
- D m/s
- E m/s

Find the velocity of when .

- A m/s
- B3 m/s
- C m/s
- D m/s
- E6 m/s