Worksheet: Rectilinear Motion with Integrals

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

Q1:

A car, starting from rest, began moving in a straight line from a fixed point. Its velocity after 𝑑 seconds is given by 𝑣 = ο€Ή 8 𝑑 + 6 𝑑  / 𝑑 β‰₯ 0 .  m s , Calculate the displacement of the car when 𝑑 = 9 s e c o n d s .

Q2:

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

  • A ο€Ή 𝑑 βˆ’ 2 7 𝑑  3 2 m
  • B ο€Ή 𝑑 βˆ’ 1 8 𝑑 + 2 0  2 m
  • C ο€Ή 𝑑 βˆ’ 1 8 𝑑  2 m
  • D ο€Ύ 𝑑 3 βˆ’ 9 𝑑 + 2 0 𝑑  3 2 m

Q3:

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

  • A ο€Ή 5 𝑑 βˆ’ 8 𝑑 + 2 0  3 2 m
  • B ( 3 0 𝑑 βˆ’ 8 ) m
  • C ( 3 0 𝑑 + 2 0 ) m
  • D ο€Ή 5 𝑑 βˆ’ 4 𝑑 + 2 0  3 2 m

Q4:

A particle moves along the π‘₯ -axis. At time 𝑑 seconds, its acceleration is given by Given that at 𝑑 = 2 s , its velocity is 28 m/s, what is its initial velocity?

Q5:

A particle moves along the positive π‘₯ axis, starting at π‘₯ = 1 m . It’s acceleration varies directly with 𝑑 2 , where 𝑑 is the time in seconds. At 𝑑 = 1 s , the particle’s displacement is 12 m, and its velocity is 14 m/s. Express the particle’s displacement, 𝑠 , and its velocity, 𝑣 , in terms of 𝑑 .

  • A 𝑣 = ο€Ό 5 0 𝑑 βˆ’ 5 8 3  / 3 m s , 𝑠 = ο€Ύ 2 5 𝑑 3 βˆ’ 5 8 𝑑 3 + 1  4 m
  • B 𝑣 = ο€Ύ 1 0 0 𝑑 3 βˆ’ 5 8 3  / 3 m s , 𝑠 = ο€Ύ 2 5 𝑑 3 βˆ’ 5 8 𝑑 3 + 1  4 m
  • C 𝑣 = ο€Ή 6 𝑑 + 1 0  / 3 m s , 𝑠 = ο€Ή 𝑑 + 1 0 𝑑 βˆ’ 1  4 m
  • D 𝑣 = ο€Ή 4 𝑑 + 1 0  / 3 m s , 𝑠 = ο€Ή 𝑑 + 1 0 𝑑 + 1  4 m
  • E 𝑣 = ο€Ή 6 𝑑 + 1 0  / 3 m s , 𝑠 = ο€Ή 𝑑 + 1 0 𝑑 + 1  4 m

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 2 𝑑 + 7 m/s2 after 𝑑 seconds of motion. If 𝑣 ( 0 ) = βˆ’ 8 m/s, how long does it take for the velocity to reach 50 m/s? Give your answer to 2 decimal places.

Q8:

If 𝑓 ( 𝑑 ) = 𝐹 ( 𝑑 ) β€² is the velocity of a particle in motion, in kilometres per hour, what is the unit of ο„Έ 𝑓 ( 𝑑 ) 𝑑 𝑏 π‘Ž d ?

  • 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 𝑑 seconds is given by Given that its initial position from a fixed point is 4 m, find an expression for its displacement at time 𝑑 seconds.

  • A ο€Ύ 5 𝑑 3 βˆ’ 𝑑 + 4  3 2 m
  • B ( 1 0 𝑑 βˆ’ 1 ) m
  • C ( 1 0 𝑑 + 4 ) m
  • D ο€Ύ 5 𝑑 3 βˆ’ 𝑑 2 + 4  3 2 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 𝑔 = 9 . 8 / m s 2 .

Q12:

The velocity function, in meters per second, for a particle moving along a line is 𝑣 ( 𝑑 ) = 3 𝑑 + 2 . Determine the particle’s displacement during the time interval 0 ≀ 𝑑 ≀ 3 .

  • A 33 m
  • B βˆ’ 3 9 2 m
  • C βˆ’ 3 3 m
  • D 3 9 2 m
  • E 2 7 2 m

Q13:

A particle started moving, from the origin, along the π‘₯ -axis. At time 𝑑 seconds, its velocity is given by 𝑣 = ο€Ή 1 . 8 𝑑 + 4 . 7 𝑑  / 𝑑 β‰₯ 0 .  m s , Find its displacement, 𝑠 , and acceleration, π‘Ž , at 𝑑 = 2 s .

  • A 𝑠 = 1 6 . 6 m , π‘Ž = 1 1 . 9 / m s 
  • B 𝑠 = 1 4 . 2 m , π‘Ž = 8 . 3 / m s 
  • C 𝑠 = 1 6 . 6 m , π‘Ž = 8 . 3 / m s 
  • D 𝑠 = 1 4 . 2 m , π‘Ž = 1 1 . 9 / m s 
  • E 𝑠 = 1 1 . 8 m , π‘Ž = 8 . 3 / m s 

Q14:

The acceleration of a particle moving in a straight line, at time 𝑑 seconds, is given by π‘Ž = ( 3 9 βˆ’ 3 𝑑 ) / 0 ≀ 𝑑 ≀ 1 3 . c m s ,  When 𝑑 > 1 3 , the particle moves with uniform velocity 𝑣 . Determine the velocity 𝑣 and the distance, 𝑑 , covered by the particle in the first 23 s of motion.

  • A 𝑣 = βˆ’ 6 9 0 / c m s , 𝑑 = 4 , 7 0 3 c m
  • B 𝑣 = 2 5 3 . 5 / c m s , 𝑑 = 4 , 2 3 2 c m
  • C 𝑣 = βˆ’ 6 9 0 / c m s , 𝑑 = 4 , 2 3 2 c m
  • D 𝑣 = 2 5 3 . 5 / c m s , 𝑑 = 4 , 7 3 2 c m
  • E 𝑣 = 7 6 0 . 5 / c m s , 𝑑 = 9 , 8 0 2 c m

Q15:

A body started moving along the π‘₯ -axis from the origin at an initial speed of 10 m/s. When it was 𝑠 meters away from the origin and moving at 𝑣 m/s, its acceleration was ( 4 5 𝑒 )   m/s2 in the direction of increasing π‘₯ . Determine 𝑠 when 𝑣 = 1 1 / m s .

  • A l n ο€Ό 6 2 3  m
  • B l n ο€Ό 2 3 3 0  m
  • C l n ο€Ό 2 3 2 0  m
  • D l n ο€Ό 3 0 2 3  m
  • E l n ο€Ό 2 0 2 3  m

Q16:

A particle is moving in a straight line such that its speed at time 𝑑 seconds is given by Given that its initial position π‘Ÿ = 1 6 0 m , find its position when 𝑑 = 3 s e c o n d s .

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/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 π‘Ž , measured in metres per second squared, and the distance π‘₯ from its starting point, measured in metres, satisfy the equation π‘Ž = π‘₯  1 5 . Find the speed 𝑣 of the particle when π‘₯ = 1 1 m .

  • A 𝑣 = 1 1 √ 5 5 1 5 m / s
  • B 𝑣 = 1 2 1 4 5 m / s
  • C 𝑣 = 2 4 2 1 5 m / s
  • D 𝑣 = 1 1 √ 1 1 0 1 5 m / s

Q19:

A body moves in a straight line. At time 𝑑 seconds, its acceleration is given by π‘Ž = ( 7 𝑑 + 1 9 ) / 𝑑 β‰₯ 0 . m s ,  Given that the initial displacement of the body is 9 m, and when 𝑑 = 2 s , its velocity is 27 m/s, what is its displacement when 𝑑 = 3 s ?

Q20:

A particle is moving in a straight line such that its velocity after 𝑑 seconds is given by Find the distance covered during the time interval between 𝑑 = 0 s and 𝑑 = 1 2 s .

Q21:

A particle is moving in a straight line such that its acceleration, π‘Ž metres per second squared, and displacement, π‘₯ meters, satisfy the equation π‘Ž = 2 6 𝑒   . Given that the particle's velocity was 12 m/s when its displacement was 0 m, find an expression for 𝑣  in terms of π‘₯ , and determine the speed 𝑣 m a x that the particle approaches as its displacement increases.

  • A 𝑣 = 1 7 0 βˆ’ 5 2 𝑒    , 𝑣 = 1 3 / m a x m s
  • B 𝑣 = 1 9 6 βˆ’ 2 6 𝑒    , 𝑣 = 1 4 / m a x m s
  • C 𝑣 = 1 7 0 βˆ’ 2 6 𝑒    , 𝑣 = 1 3 / m a x m s
  • D 𝑣 = 1 9 6 βˆ’ 5 2 𝑒    , 𝑣 = 1 4 / m a x m s

Q22:

A particle is moving in a straight line such that its velocity at time 𝑑 seconds is given by 𝑣 = [ βˆ’ ( 4 𝑑 ) + 1 4 ] / 𝑑 β‰₯ 0 . s i n m s , Given that its initial position π‘Ÿ = 1 3  m , find an expression for its position at time 𝑑 seconds.

  • A  1 4 ( 4 𝑑 ) + 5 1 4  c o s m
  • B [ βˆ’ 4 ( 4 𝑑 ) + 1 3 ] c o s m
  • C [ βˆ’ 4 ( 4 𝑑 ) + 1 7 ] c o s m
  • D  1 4 𝑑 + 1 4 ( 4 𝑑 ) + 5 1 4  c o s 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 𝑑 = 7 s ?

Q25:

A particle started moving in a straight line. Its acceleration at time 𝑑 seconds is given by π‘Ž = ο€Ή βˆ’ 5 𝑑 + 5  / , 𝑑 β‰₯ 0 .   m s Find the maximum velocity of the particle 𝑣 m a x and the distance π‘₯ it travelled before it attained this velocity.

  • A 𝑣 = 2 0 3 / m a x m s , π‘₯ = 2 5 1 2 m
  • B 𝑣 = 1 0 3 / m a x m s , π‘₯ = 1 0 3 m
  • C 𝑣 = 2 0 3 / m a x m s , π‘₯ = 1 0 3 m
  • D 𝑣 = 1 0 3 / m a x m s , π‘₯ = 2 5 1 2 m

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.