# 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 particle’s displacement relative to the origin was m in the direction of increasing , its velocity was . Determine the particle’s 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 particle’s 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 particle’s 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 particle’s 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 particle’s 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 particle’s 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 particle’s velocity, , and acceleration, , at .

• A,
• B,
• C,
• D,
• E,

Q14:

A particle is moving in a straight line. After time seconds, where , the body’s 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 particle’s 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:

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