Worksheet: Linear Motion with Derivatives

In this worksheet, we will practice using differentiation to get the average and instantaneous velocities and acceleration vectors of a particle in straight-line motion.

Q1:

A particle started moving along the π‘₯ -axis. When the particle’s displacement relative to the origin was ⃑ 𝑠 m in the direction of increasing π‘₯ , its velocity was ⃑ 𝑣 = 5 ⃑ 𝑠 / m s . Determine the particle’s acceleration when ⃑ 𝑠 = 1 3 m .

Q2:

A particle started moving along the π‘₯ -axis. At time 𝑑 seconds, its position relative to the origin is given by π‘₯ = [ 1 2 ( 1 4 𝑑 ) + 9 ( 1 4 𝑑 ) ] , 𝑑 β‰₯ 0 . s i n c o s m Find the maximum distance between the particle and the origin π‘₯ m a x , and determine the velocity of the particle 𝑣 when 𝑑 = 3 πœ‹ s .

  • A π‘₯ = 7 2 5 m a x m , 𝑣 = 1 6 8 / m s
  • B π‘₯ = 1 5 m a x m , 𝑣 = βˆ’ 1 6 8 / m s
  • C π‘₯ = 7 2 5 m a x m , 𝑣 = βˆ’ 1 6 8 / m s
  • D π‘₯ = 1 5 m a x m , 𝑣 = 1 6 8 / m s

Q3:

A particle moves along the π‘₯ -axis such that at time 𝑑 seconds its displacement from the origin is given by π‘₯ = ( 1 0 2 𝑑 ) 𝑑 β‰₯ 0 . s i n m , Determine the particle’s acceleration when π‘₯ = βˆ’ 5 m .

Q4:

A particle moves along the π‘₯ -axis so that its position relative to the origin after time 𝑑 (where 𝑑 β‰₯ 0 ) is given by What is the particle’s maximum displacement?

Q5:

A stone is projected vertically upwards. At time 𝑑 seconds, its height from the ground is given by β„Ž = ο€Ή 4 6 . 6 𝑑 βˆ’ 4 . 9 𝑑  𝑑 β‰₯ 0 .  m , Determine the speed of the stone when it is 22.5 m high.

Q6:

A particle is moving in a straight line such that its velocity 𝑣 and position π‘₯ satisfy the following equation: 𝑣 = 1 6 ( 5 4 βˆ’ π‘₯ ) .   Find an expression for the particle’s acceleration in terms of π‘₯ .

  • A βˆ’ 1 1 2 π‘₯
  • B π‘₯ 3 ( π‘₯ βˆ’ 5 4 )  
  • C βˆ’ π‘₯ 2 π‘₯ βˆ’ 5 4
  • D π‘₯ 6 ( π‘₯ βˆ’ 5 4 )  

Q7:

A particle is moving in a straight line such that its displacement 𝑠 after 𝑑 seconds is given by 𝑠 = ο€Ή βˆ’ 1 0 𝑑  + 1 2 𝑑  + 1 0 𝑑  m , 𝑑 β‰₯ 0 . Find the velocity of the particle when its acceleration is zero.

Q8:

A particle is moving in a straight line such that its displacement π‘₯ at time 𝑑 seconds is given by π‘₯ = ο€Ή 9 . 6 𝑑 βˆ’ 𝑑  𝑑 β‰₯ 0 .  m , What distance does the particle travel in the first 9.6 seconds?

Q9:

A particle is moving in a straight line such that its displacement π‘₯ after 𝑑 seconds is given by π‘₯ = ο€Ή 2 𝑑 βˆ’ 2 4 𝑑 βˆ’ 1 5  𝑑 β‰₯ 0 .  m , Determine the time after which the particle changes its direction.

Q10:

A particle moving along a path has velocity 𝑣 and acceleration π‘Ž . Given that the equation of the displacement is π‘₯ = 𝑑 t a n , find π‘Ž .

  • A 𝑣 π‘₯
  • B βˆ’ 2 𝑣 π‘₯
  • C βˆ’ 𝑣 π‘₯
  • D 2 𝑣 π‘₯

Q11:

A particle moves along the π‘₯ -axis. At time 𝑑 seconds, its displacement from the origin is given by π‘₯ = ο€Ή 2 𝑑 βˆ’ 6 𝑑 βˆ’ 4  𝑑 β‰₯ 0 .  m , Determine all the possible values of 𝑑 , in seconds, at which the particle’s speed β€– 𝑣 β€– = 4 m/s.

  • A2, 10
  • B1, 5
  • C 1 5 , 1
  • D 1 2 , 5 2
  • E 2 5 , 2

Q12:

A particle moves along the π‘₯ -axis. When its displacement from the origin is 𝑠 m, its velocity is given by 𝑣 = ο„ž 1 βˆ’ 1 4 𝑠 + 9 8 / .  m s Find the particle’s minimum velocity.

  • A √ 7 4 9 m/s
  • B √ 7 7 m/s
  • C √ 7 9 8 m/s
  • D √ 2 1 4 m/s
  • E √ 2 2 8 m/s

Q13:

A particle moves along the π‘₯ -axis such that at time 𝑑 seconds its displacement from the origin is given by π‘₯ = [ 1 5 ( 2 𝑑 ) + 1 0 ( 2 𝑑 ) + 1 1 8 ] 𝑑 β‰₯ 0 . s i n c o s m , Find the particle’s velocity, 𝑣 , and acceleration, π‘Ž , at 𝑑 = πœ‹ s .

  • A 𝑣 = βˆ’ 3 2 . 3 2 / m s , π‘Ž = βˆ’ 3 1 . 9 6 / m s 
  • B 𝑣 = βˆ’ 3 0 / m s , π‘Ž = 4 0 / m s 
  • C 𝑣 = βˆ’ 3 0 / m s , π‘Ž = βˆ’ 3 1 . 9 6 / m s 
  • D 𝑣 = 3 0 / m s , π‘Ž = βˆ’ 4 0 / m s 
  • E 𝑣 = 3 0 / m s , π‘Ž = 4 0 / m s 

Q14:

A particle is moving in a straight line. After time 𝑑 seconds, where 𝑑 β‰₯ 0 , the body’s displacement relative to a fixed point is given by ⃑ 𝑠 =  ο€Ό 5 6 𝑑 + 5 𝑑  ⃑ 𝑐   m , where ⃑ 𝑐 is a fixed unit vector. Find the initial velocity of the particle ⃑ 𝑣  and its acceleration ⃑ π‘Ž , 5 seconds after it started moving.

  • A ⃑ 𝑣 = ο€Ή 5 ⃑ 𝑐  /  m s , ⃑ π‘Ž = ο€Ή 3 0 ⃑ 𝑐  / m s 
  • B ⃑ 𝑣 = ο€Ό ο€Ό 5 2 𝑑 + 5  ⃑ 𝑐  /   m s , ⃑ π‘Ž = ο€Ή 2 5 ⃑ 𝑐  / m s 
  • C ⃑ 𝑣 = ο€Ό ο€Ό 5 2 𝑑 + 5  ⃑ 𝑐  /   m s , ⃑ π‘Ž = ο€Ή 3 0 ⃑ 𝑐  / m s 
  • D ⃑ 𝑣 = ο€Ή 5 ⃑ 𝑐  /  m s , ⃑ π‘Ž = ο€Ή 2 5 ⃑ 𝑐  / m s 

Q15:

A particle moves along the π‘₯ -axis such that at time 𝑑 seconds its velocity is given by 𝑣 = ο€Ή 𝑑 βˆ’ 1 2 𝑑 + 3  / 𝑑 β‰₯ 0 .  m s , After how many seconds is its acceleration equal to 0?

Q16:

A particle moves along a straight line. Its displacement at time 𝑑 is π‘₯ = βˆ’ ( 𝑑 ) c o s . Which of the following statements about the acceleration of the particle is true?

  • A it is equal to the velocity of the particle
  • B it is equal to π‘₯
  • C it is equal to βˆ’ 𝑣 , where 𝑣 is the velocity of the particle
  • D it is equal to βˆ’ π‘₯

Q17:

A particle is moving in a straight line such that its speed 𝑣 , measured in metres per second, and its position π‘₯ , measured in metres, satisfy the following equation: 𝑣 = 3 3 βˆ’ 3 π‘₯ .  c o s Find the maximum speed of the particle 𝑣 m a x and the acceleration of the particle π‘Ž when 𝑣 = 𝑣 m a x .

  • A 𝑣 = 5 . 4 8 / m a x m s , π‘Ž = 0 / m s 
  • B 𝑣 = 5 . 4 8 / m a x m s , π‘Ž = 1 . 5 / m s 
  • C 𝑣 = 6 / m a x m s , π‘Ž = 1 . 5 / m s 
  • D 𝑣 = 6 / m a x m s , π‘Ž = 0 / m s 

Q18:

A particle is moving in a straight line. The relation between its velocity 𝑣 , measured in metres per second, and its position π‘₯ , measured in metres, is given by 𝑣 = 7 ο€Ή 1 0 0 βˆ’ π‘₯  .   Find the magnitude of its acceleration when its velocity is zero.

Q19:

A particle is moving in a straight line such that its velocity 𝑣 at time 𝑑 seconds is given by 𝑣 = ο€Ή 2 𝑑 βˆ’ 6 8  / 𝑑 β‰₯ 0 .  m s , Find the magnitude of the acceleration of the particle when its velocity is 94 m/s.

Q20:

A particle moves along a straight line. Its displacement at time 𝑑 is π‘₯ = ( 𝑑 ) c o s . Which of the following statements about the acceleration of the particle is true?

  • A it is equal to the velocity of the particle
  • B it is equal to π‘₯
  • C it is equal to βˆ’ 𝑣 , where 𝑣 is the velocity of the particle
  • D it is equal to βˆ’ π‘₯

Q21:

A particle moves along a straight line. Its displacement at time 𝑑 is π‘₯ = 𝑑 s i n . Which of the following statements about the acceleration of the particle is true?

  • A it is equal to the velocity of the particle
  • B it is equal to π‘₯
  • C it is equal to βˆ’ 𝑣 , where 𝑣 is the velocity of the particle
  • D it is equal to βˆ’ π‘₯

Q22:

A particle is moving in a straight line such that its displacement from the origin after 𝑑 seconds is given by π‘₯ = ο€Ό 1 3 2 𝑑  𝑑 β‰₯ 0 . c o s m , Find its velocity 𝑣 when 𝑑 = πœ‹ 4 s and its acceleration π‘Ž when 𝑑 = πœ‹ 3 s .

  • A 𝑣 = 4 3 / m s , π‘Ž = βˆ’ 4 3 / m s 
  • B 𝑣 = βˆ’ 2 3 / m s , π‘Ž = βˆ’ 2 3 / m s 
  • C 𝑣 = βˆ’ 4 3 / m s , π‘Ž = 4 3 / m s 
  • D 𝑣 = βˆ’ 2 3 / m s , π‘Ž = 2 3 / m s 

Q23:

A particle moves along the π‘₯ -axis. At time 𝑑 seconds, its displacement from the origin is given by π‘₯ = ο€Ή π‘Ž 𝑑 βˆ’ 𝑑 + 𝑏  𝑑 β‰₯ 0 .  m , When 𝑑 = 1 s , π‘₯ = 7 m , and when 𝑑 = 2 s , the particle’s velocity is 7 m/s. Determine the value of 𝑏 βˆ’ π‘Ž .

Q24:

A particle moves along a straight line. Its displacement at time 𝑑 is π‘₯ = βˆ’ 𝑑 t a n . Find its velocity, 𝑣 , and hence determine which of the following expressions is equal to the acceleration of the particle.

  • A 𝑣 π‘₯
  • B 2 𝑣 π‘₯
  • C βˆ’ 𝑣 π‘₯
  • D βˆ’ 2 𝑣 π‘₯

Q25:

A particle moves along a straight line. Its displacement at time 𝑑 is π‘₯ = βˆ’ ( 𝑑 ) s i n . Which of the following statements about the acceleration of the particle is true?

  • AIt is equal to the velocity of the particle.
  • BIt is equal to π‘₯ .
  • CIt is equal to βˆ’ 𝑣 , where 𝑣 is the velocity of the particle.
  • DIt is equal to βˆ’ π‘₯ .

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