# Worksheet: Rectilinear Motion and Integration

In this worksheet, we will practice applying integrals to solve problems involving motion in a straight line.

Q1:

A car, starting from rest, began moving in a straight line from a fixed point. Its velocity after seconds is given by Calculate the displacement of the car when .

Q2:

A particle is moving in a straight line such that its acceleration at time seconds is given by Given that its initial velocity is 20 m/s, find an expression for its displacement in terms of .

• A m
• B m
• C m
• D m

Q3:

A particle is moving in a straight line such that its velocity at time seconds is given by Given that its initial position from a fixed point is 20 m, find an expression for its displacement at time seconds.

• A m
• B m
• C m
• D m

Q4:

A particle moves along the -axis. At time seconds, its acceleration is given by Given that at , its velocity is 28 m/s, what is its initial velocity?

Q5:

A particle moves along the positive -axis, starting at . Its acceleration varies directly with , where is the time in seconds. At , the particle’s displacement is 12 m, and its velocity is 14 m/s. Express the particle’s displacement, , and its velocity, , in terms of .

• A ,
• B ,
• C ,
• D ,
• E ,

Q6:

The figure shows a velocity-time graph for a particle moving in a straight line. Find the total distance the particle travelled.

Q7:

A particle accelerates at the rate of m/s2 after seconds of motion. If m/s, how long does it take for the velocity to reach 50 m/s? Give your answer to 2 decimal places.

Q8:

If is the velocity of a particle in motion, in kilometers per hour, what is the unit of ?

• Akilometers
• Bhours per kilometer
• Ckilometers per hour
• Dhours

Q9:

A particle is moving in a straight line such that its velocity at time seconds is given by Given that its initial position from a fixed point is 4 m, find an expression for its displacement at time seconds.

• A m
• B m
• C m
• D m

Q10:

The figure shows a velocity-time graph for a particle moving in a straight line. Find the total distance the particle travelled.

Q11:

A sky diver jumped out of a plane. His terminal velocity was 55 m/s. Given that the air resistance is directly proportional to the velocity, how long did it take for his speed to reach 54 m/s? Round your answer to three significant figures. Take .

Q12:

The velocity function, in meters per second, for a particle moving along a line is . Determine the particle’s displacement during the time interval .

• A m
• B m
• C33 m
• D m
• E m

Q13:

A particle started moving, from the origin, along the -axis. At time seconds, its velocity is given by Find its displacement, , and acceleration, , at .

• A ,
• B ,
• C ,
• D ,
• E ,

Q14:

The acceleration of a particle moving in a straight line, at time seconds, is given by When , the particle moves with uniform velocity . Determine the velocity and the distance, , covered by the particle in the first 23 s of motion.

• A ,
• B ,
• C ,
• D ,
• E ,

Q15:

A body started moving along the -axis from the origin at an initial speed of 10 m/s. When it was meters away from the origin and moving at m/s, its acceleration was m/s2 in the direction of increasing . Determine when .

• A m
• B m
• C m
• D m
• E m

Q16:

A particle is moving in a straight line such that its speed at time seconds is given by Given that its initial position , find its position when .

Q17:

A particle started moving in a straight line from point toward point . Its velocity after seconds is given by . After 2 seconds, another particle started moving from rest in a straight line from point toward point . This particle was accelerating at 0.9 m/s2. The two particles collided 6 seconds after the first particle started moving. Find the distance .

Q18:

A particle, starting from rest, began moving in a straight line. Its acceleration , measured in meters per second squared, and the distance from its starting point, measured in meters, satisfy the equation . Find the speed of the particle when .

• A
• B
• C
• D

Q19:

A body moves in a straight line. At time seconds, its acceleration is given by Given that the initial displacement of the body is 9 m, and when , its velocity is 27 m/s, what is its displacement when ?

Q20:

A particle is moving in a straight line such that its velocity after seconds is given by Find the distance covered during the time interval between and .

Q21:

A particle is moving in a straight line such that its acceleration, meters per second squared, and displacement, meters, satisfy the equation . Given that the particle's velocity was 12 m/s when its displacement was 0 m, find an expression for in terms of , and determine the speed that the particle approaches as its displacement increases.

• A ,
• B ,
• C ,
• D ,

Q22:

A particle is moving in a straight line such that its velocity at time seconds is given by Given that its initial position , find an expression for its position at time seconds.

• A m
• B m
• C m
• D m

Q23:

The figure shows a velocity-time graph for a particle moving in a straight line. Find the magnitude of the displacement of the particle.

Q24:

The diagram below shows the acceleration of a particle which was initially at rest. What was its velocity at ?

Q25:

A particle started moving, from rest, in a straight line. Its acceleration at time seconds after it started moving is given by . Find the maximum velocity of the particle () and the distance it traveled from its starting position before it attained this velocity.

• A ,
• B ,
• C ,
• D ,