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 𝑑 seconds is given by 𝑣=ο€Ή8𝑑+6𝑑/𝑑β‰₯0.ms, Calculate the displacement of the car when 𝑑=9seconds.

Q2:

A particle is moving in a straight line such that its acceleration at time 𝑑 seconds is given by π‘Ž=(2π‘‘βˆ’18)/𝑑β‰₯0.ms, Given that its initial velocity is 20 m/s, find an expression for its displacement in terms of 𝑑.

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

Q3:

A particle is moving in a straight line such that its velocity at time 𝑑 seconds is given by 𝑣=ο€Ή15π‘‘βˆ’8𝑑/𝑑β‰₯0.ms, 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    m
  • B ( 3 0 𝑑 βˆ’ 8 ) m
  • C ο€Ή 5 𝑑 βˆ’ 4 𝑑 + 2 0    m
  • D ( 3 0 𝑑 + 2 0 ) m

Q4:

A particle moves along the π‘₯-axis. At time 𝑑 seconds, its acceleration is given by π‘Ž=(4𝑑+6)/𝑑β‰₯0.ms, Given that at 𝑑=2s, its velocity is 28 m/s, what is its initial velocity?

Q5:

A particle moves along the positive π‘₯-axis, starting at π‘₯=1m. Its acceleration varies directly with π‘‘οŠ¨, where 𝑑 is the time in seconds. At 𝑑=1s, 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  /  m s , 𝑠 = ο€Ύ 2 5 𝑑 3 βˆ’ 5 8 𝑑 3 + 1  οŠͺ m
  • B 𝑣 = ο€Ύ 1 0 0 𝑑 3 βˆ’ 5 8 3  /  m s , 𝑠 = ο€Ύ 2 5 𝑑 3 βˆ’ 5 8 𝑑 3 + 1  οŠͺ m
  • C 𝑣 = ο€Ή 6 𝑑 + 1 0  /  m s , 𝑠 = ο€Ή 𝑑 + 1 0 𝑑 βˆ’ 1  οŠͺ m
  • D 𝑣 = ο€Ή 6 𝑑 + 1 0  /  m s , 𝑠 = ο€Ή 𝑑 + 1 0 𝑑 + 1  οŠͺ m
  • E 𝑣 = ο€Ή 4 𝑑 + 1 0  /  m s , 𝑠 = ο€Ή 𝑑 + 1 0 𝑑 + 1  οŠͺ 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 2𝑑+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) is the velocity of a particle in motion, in kilometers per hour, what is the unit of 𝑓(𝑑)π‘‘οŒ»οŒΊd?

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

Q9:

A particle is moving in a straight line such that its velocity at time 𝑑 seconds is given by 𝑣=ο€Ή5π‘‘βˆ’π‘‘ο…/𝑑β‰₯0.ms, 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    m
  • B ( 1 0 𝑑 βˆ’ 1 ) m
  • C ο€Ύ 5 𝑑 3 βˆ’ 𝑑 2 + 4    m
  • D ( 1 0 𝑑 + 4 ) 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/ms.

Q12:

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

  • A 3 9 2 m
  • B βˆ’ 3 9 2 m
  • C33 m
  • D 2 7 2 m
  • E βˆ’ 3 3 m

Q13:

A particle started moving, from the origin, along the π‘₯-axis. At time 𝑑 seconds, its velocity is given by 𝑣=ο€Ή1.8𝑑+4.7𝑑/𝑑β‰₯0.ms, Find its displacement, 𝑠, and acceleration, π‘Ž, at 𝑑=2s.

  • A 𝑠 = 1 4 . 2 m , π‘Ž = 1 1 . 9 / m s 
  • B 𝑠 = 1 6 . 6 m , π‘Ž = 1 1 . 9 / m s 
  • C 𝑠 = 1 6 . 6 m , π‘Ž = 8 . 3 / m s 
  • D 𝑠 = 1 4 . 2 m , π‘Ž = 8 . 3 / 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=(39βˆ’3𝑑)/0≀𝑑≀13.cms, When 𝑑>13, 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 = βˆ’ 6 9 0 / c m s , 𝑑 = 4 , 2 3 2 c m
  • C 𝑣 = 2 5 3 . 5 / c m s , 𝑑 = 4 , 7 3 2 c m
  • D 𝑣 = 7 6 0 . 5 / c m s , 𝑑 = 9 , 8 0 2 c m
  • E 𝑣 = 2 5 3 . 5 / c m s , 𝑑 = 4 , 2 3 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 (45𝑒) m/s2 in the direction of increasing π‘₯. Determine 𝑠 when 𝑣=11/ms.

  • A l n ο€Ό 2 3 3 0  m
  • B l n ο€Ό 6 2 3  m
  • C l n ο€Ό 3 0 2 3  m
  • D l n ο€Ό 2 3 2 0  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 𝑣=(10𝑑+2)/𝑑β‰₯0.ms, Given that its initial position π‘Ÿ=16m, find its position when 𝑑=3seconds.

Q17:

A particle started moving in a straight line from point 𝐴 toward point 𝐡. Its velocity after 𝑑 seconds is given by 𝑣=ο€Ή0.7𝑑+0.2𝑑/𝑑β‰₯0ms,. After 2 seconds, another particle started moving from rest in a straight line from point 𝐡 toward point 𝐴. This particle was accelerating at 0.9 m/s2. The two particles collided 6 seconds after the first particle started moving. Find the distance 𝐴𝐡.

Q18:

A particle, starting from rest, began moving in a straight line. Its acceleration π‘Ž, measured in meters per second squared, and the distance π‘₯ from its starting point, measured in meters, satisfy the equation π‘Ž=π‘₯15. Find the speed 𝑣 of the particle when π‘₯=11m.

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

Q19:

A body moves in a straight line. At time 𝑑 seconds, its acceleration is given by π‘Ž=(7𝑑+19)/𝑑β‰₯0.ms, Given that the initial displacement of the body is 9 m, and when 𝑑=2s, its velocity is 27 m/s, what is its displacement when 𝑑=3s?

Q20:

A particle is moving in a straight line such that its velocity after 𝑑 seconds is given by 𝑣=ο€Ήπ‘‘βˆ’12𝑑+20𝑑/𝑑β‰₯0.ms, Find the distance covered during the time interval between 𝑑=0s and 𝑑=12s.

Q21:

A particle is moving in a straight line such that its acceleration, π‘Ž meters per second squared, and displacement, π‘₯ meters, satisfy the equation π‘Ž=26π‘’οŠ±ο—. Given that the particle's velocity was 12 m/s when its displacement was 0 m, find an expression for π‘£οŠ¨ in terms of π‘₯, and determine the speed 𝑣max that the particle approaches as its displacement increases.

  • A 𝑣 = 1 9 6 βˆ’ 2 6 𝑒    , 𝑣 = 1 4 / m a x m s
  • B 𝑣 = 1 7 0 βˆ’ 2 6 𝑒    , 𝑣 = 1 3 / m a x m s
  • C 𝑣 = 1 7 0 βˆ’ 5 2 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’(4𝑑)+14]/𝑑β‰₯0.sinms, Given that its initial position π‘Ÿ=13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  1 4 𝑑 + 1 4 ( 4 𝑑 ) + 5 1 4  c o s m
  • D [ βˆ’ 4 ( 4 𝑑 ) + 1 7 ] 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=7s?

Q25:

A particle started moving, from rest, in a straight line. Its acceleration at time 𝑑 seconds after it started moving is given by π‘Ž=ο€Ήβˆ’5𝑑+5/,𝑑β‰₯0ms. Find the maximum velocity of the particle (𝑣max) and the distance π‘₯ it traveled from its starting position 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 = 1 0 3 / m a x m s , π‘₯ = 2 5 1 2 m
  • D 𝑣 = 2 0 3 / m a x m s , π‘₯ = 1 0 3 m

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