In this worksheet, we will practice using definite integrals to solve problems involving rectilinear motion.

**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 . It’s 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/s^{2} 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 kilometres per hour, what is the unit of ?

- A hours
- B kilometres per hour
- C hours per kilometre
- D kilometres

**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 33 m
- B m
- C 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/s^{2}
in the direction of increasing .
Determine when .

- A m
- B m
- C m
- D m
- E m

**Q17: **

A particle started moving in a straight line from point towards point .
Its velocity after seconds is given by
After 2 seconds, another particle started moving in a straight line from point towards point .
This particle was accelerating at 0.9 m/s^{2}.
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 metres per second squared, and the distance from its starting point, measured in metres, satisfy the equation . Find the speed of the particle when .

- A
- B
- C
- D

**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, metres 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 in a straight line. Its acceleration at time seconds is given by Find the maximum velocity of the particle and the distance it travelled before it attained this velocity.

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