# Worksheet: Linear Motion with Derivatives

Q1:

A particle is moving in a straight line. After time seconds, where , the bodyโ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 ,

Q2:

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

Q3:

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

Q4:

A particle moves along the -axis. When its displacement from the origin is m, its velocity is given by Find the particleโs minimum velocity.

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

Q5:

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?

Q6:

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?

• A it is equal to the velocity of the particle
• B it is equal to
• C it is equal to , where is the velocity of the particle
• D it is equal to

Q7:

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 ,

Q8:

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 following equation: Find the maximum speed of the particle and the acceleration of the particle when .

• A ,
• B ,
• C ,
• D ,

Q9:

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.

Q10:

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.

• Aundefined
• B 10 m/s2
• C 0 m/s2
• D 70 m/s2

Q11:

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.

Q12:

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?

• A it is equal to the velocity of the particle
• B it is equal to
• C it is equal to , where is the velocity of the particle
• D it is equal to

Q13:

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?

• A it is equal to the velocity of the particle
• B it is equal to
• C it is equal to , where is the velocity of the particle
• D it is equal to

Q14:

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.

Q15:

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?

Q16:

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 particleโs speed m/s.

• A2, 10
• B1, 5
• C , 1
• D ,
• E , 2

Q17:

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 ,

Q18:

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

Q19:

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

Q20:

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

• A
• B
• C
• D

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

Q22:

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

• A ,
• B ,
• C ,
• D ,
• E ,

Q23:

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

• A
• B
• C
• D

Q24:

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

Q25:

A particle moves in a straight line such that at time seconds, its velocity is given by After how many seconds does the direction of the particleโs motion change?