# 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 particle’s 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/s2 and the velocity m/s is given by , where is a constant. The initial acceleration of P is 10 m/s2.

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/s2
• BMaximum velocity = 2 m/s, maximum acceleration = 10 m/s2
• CMaximum velocity = 10 m/s, maximum acceleration = 6 m/s2
• DMaximum velocity = 6 m/s, maximum acceleration = 0 m/s2
• EMaximum velocity = 10 m/s, maximum acceleration = 2 m/s2

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/s2
• B m/s2
• C m/s2
• D m/s2
• E m/s2

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

This lesson includes 81 additional question variations for subscribers.