# Worksheet: Linear Motion with Derivatives

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

**Q1: **

A particle started moving along the -axis. When the particleβs displacement relative to the origin was m in the direction of increasing , its velocity was . Determine the particleβs acceleration when .

**Q2: **

A particle started moving along the -axis. At time seconds, its position relative to the origin is given by Find the maximum distance between the particle and the origin , and determine the velocity of the particle when .

- A ,
- B ,
- C ,
- D ,

**Q3: **

A particle moves along the -axis such that at time seconds its displacement from the origin is given by Determine the particleβs acceleration when .

**Q4: **

A particle moves along the -axis so that its position relative to the origin after time (where ) is given by What is the particleβs maximum displacement?

**Q5: **

A stone is projected vertically upward. At time seconds, its height from the ground is given by Determine the speed of the stone when it is 22.5 m high.

**Q6: **

A particle is moving in a straight line such that its velocity and position satisfy the following equation: Find an expression for the particleβs acceleration in terms of .

- A
- B
- C
- D

**Q7: **

A particle is moving in a straight line such that its displacement after seconds is given by Find the velocity of the particle when its acceleration is zero.

**Q8: **

A particle is moving in a straight line such that its displacement at time seconds is given by What distance does the particle travel in the first 9.6 seconds?

**Q9: **

A particle is moving in a straight line such that its displacement after seconds is given by Determine the time after which the particle changes its direction.

**Q10: **

A particle moving along a path has velocity and acceleration . Given that the equation of the displacement is , find .

- A
- B
- C
- D

**Q11: **

A particle moves along the -axis. At time seconds, its displacement from the origin is given by Determine all the possible values of , in seconds, at which the particleβs speed m/s.

- A2, 10
- B , 2
- C1, 5
- D , 1
- E ,

**Q12: **

A particle moves along the . When its displacement from the origin is m, its velocity is given by Find the particleβs minimum velocity.

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

**Q13: **

A particle moves along the -axis such that at time seconds its displacement from the origin is given by Find the particleβs velocity, , and acceleration, , at .

- A ,
- B ,
- C ,
- D ,
- E ,

**Q14: **

A particle is moving in a straight line. After time seconds, where , the bodyβs displacement relative to a fixed point is given by , where is a fixed unit vector. Find the initial velocity of the particle and its acceleration , 5 seconds after it started moving.

- A ,
- B ,
- C ,
- D ,

**Q15: **

A particle moves along the -axis such that at time seconds its velocity is given by After how many seconds is its acceleration equal to 0?

**Q16: **

A particle moves along a straight line. Its displacement at time is . Which of the following statements about the acceleration of the particle is true?

- Ait is equal to
- Bit is equal to , where is the velocity of the particle
- Cit is equal to the velocity of the particle
- Dit is equal to

**Q17: **

A particle is moving in a straight line such that its speed , measured in meters per second, and its position , measured in meters, satisfy the equation . Find the maximum speed of the particle and the acceleration of the particle when .

- A ,
- B ,
- C ,
- D ,

**Q18: **

A particle is moving in a straight line. The relation between its velocity , measured in meters per second, and its position , measured in meters, is given by . Find the magnitude of its acceleration when its velocity is zero.

**Q19: **

A particle is moving in a straight line such that its velocity at time seconds is given by Find the magnitude of the acceleration of the particle when its velocity is 94 m/s.

**Q20: **

A particle moves along a straight line. Its displacement at time is . Which of the following statements about the acceleration of the particle is true?

- Ait is equal to
- Bit is equal to
- Cit is equal to , where is the velocity of the particle
- Dit is equal to the velocity of the particle

**Q21: **

A particle moves along a straight line. Its displacement at time is . Which of the following statements about the acceleration of the particle is true?

- Ait is equal to
- Bit is equal to the velocity of the particle
- Cit is equal to , where is the velocity of the particle
- Dit is equal to

**Q22: **

A particle is moving in a straight line such that its displacement from the origin after seconds is given by Find its velocity when and its acceleration when .

- A ,
- B ,
- C ,
- D ,

**Q23: **

A particle moves along the -axis. At time seconds, its displacement from the origin is given by When , , and when , the particleβs velocity is 7 m/s. Determine the value of .

**Q24: **

A particle moves along a straight line. Its displacement at time is . Find its velocity, , and hence determine which of the following expressions is equal to the acceleration of the particle.

- A
- B
- C
- D

**Q25: **

- AIt is equal to the velocity of the particle.
- BIt is equal to .
- CIt is equal to .
- DIt is equal to , where is the velocity of the particle.