Worksheet: Applications of Derivatives on Rectilinear Motion

In this worksheet, we will practice applying derivatives to problems of motion in a straight line.

Q1:

A particle is moving in a straight line such that its displacement 𝑠 after 𝑑 seconds is given by 𝑠 = ο€Ή 3 𝑑 βˆ’ 5 4 𝑑 + 3 8 𝑑  𝑑 β‰₯ 0 .   m , Determine the time interval during which the velocity of the particle is increasing.

  • A ( 1 8 , ∞ )
  • B ( 1 2 , ∞ )
  • C ( 3 6 , ∞ )
  • D ( 6 , ∞ )

Q2:

A particle is moving in a straight line such that its position π‘Ÿ meters relative to the origin at time 𝑑 seconds is given by π‘Ÿ = ο€Ή 𝑑  + 3 𝑑 + 7  . Find the particle’s average velocity between 𝑑 = 2 s and 𝑑 = 4 s .

  • A 18 m/s
  • B 11 m/s
  • C βˆ’ 2 m/s
  • D 9 m/s

Q3:

A particle started moving along the π‘₯ -axis. At time 𝑑 seconds, the its displacement from the origin is given by π‘₯ = ο€Ή 𝑑 βˆ’ 3 𝑑  𝑑 β‰₯ 0 .   m , Find the body’s average speed within the time interval [ 0 , 5 ] .

Q4:

A particle moves along the π‘₯ -axis such that at time 𝑑 seconds, its displacement from the origin is given by 𝑠 = ο€Ή 𝑑 βˆ’ 4 𝑑  𝑑 β‰₯ 0 .   m , What is the particle’s average velocity in the first 10 seconds?

Q5:

A particle is moving in a straight line such that its displacement 𝑠 in metres is given as a function of time 𝑑 in seconds by 𝑠 = 5 𝑑  βˆ’ 8 4 𝑑  + 3 3 𝑑 , 𝑑 β‰₯ 0 . Find the magnitude of the acceleration of the particle when the velocity is zero.

Q6:

A particle moves along the π‘₯ -axis. At time 𝑑 seconds, its displacement from the origin is given by 𝑠 = ο€Ή βˆ’ 𝑑  + 4  m , 𝑑 β‰₯ 0 . Determine the time at which the particle’s acceleration is 9 m/s2.

  • A 1 5 7 s
  • B 3 s
  • C 4 1 2 s
  • D 1 1 2 s
  • E 2 1 2 s

Q7:

A particle moves in a straight line such that at time 𝑑 seconds its displacement from a fixed point on the line is given by 𝑠 = ο€Ή 𝑑 βˆ’ 𝑑 βˆ’ 3  𝑑 β‰₯ 0 .   m , Determine whether the particle is accelerating or decelerating when 𝑑 = 2 s .

  • Aaccelerating
  • Bdecelerating

Q8:

A body moves along the π‘₯ -axis such that at time 𝑑 seconds, its displacement from the origin is given by 𝑠 = ο€Ή 6 𝑑 οŠͺ βˆ’ 𝑑  βˆ’ 3 𝑑  βˆ’ 4 𝑑 + 3  m , 𝑑 β‰₯ 0 . What is its velocity when its acceleration is equal to 0?

  • A βˆ’ 1 3 9 m/s
  • B βˆ’ 6 9 1 3 m/s
  • C βˆ’ 4 4 9 m/s
  • D βˆ’ 4 9 9 m/s
  • E βˆ’ 1 3 8 m/s

Q9:

A particle is moving in a straight line such that its displacement 𝑠 at 𝑑 seconds is given by 𝑠 = ο€Ή 5 𝑑 βˆ’ 3 0 𝑑 + 4 𝑑  𝑑 β‰₯ 0 .   m , Find the velocity of the particle when the acceleration is zero.

Q10:

A particle is moving in a straight line such that its displacement in meters, 𝑠 , after 𝑑 seconds is given by 𝑠 = βˆ’ 1 2 𝑑  + 1 2 𝑑  βˆ’ 3 𝑑 m . When the particle’s velocity is zero, its acceleration is π‘Ž m/s2. Find all the possible values of π‘Ž .

  • A24, 22
  • B27, 25
  • C12, 10
  • D βˆ’ 1 2 , 12

Q11:

A particle moves along the π‘₯ -axis such that at time seconds ( 𝑑 β‰₯ 0 ) its velocity is given by 𝑣 = ( 3 𝑑 βˆ’ 9 𝑑 ) / 2 m s . Determine the time interval in which the particle decelerates.

  • A  0 , 3 2 
  • B ] 0 , 3 ]
  • C ] 3 , ∞ [
  • D  3 2 , 3 
  • E  βˆ’ ∞ , 3 2 

Q12:

A particle moves along the π‘₯ -axis. When its displacement from the origin is 𝑠 m, its velocity is given by 𝑣 = 4 3 + 𝑠 m / s . Find the particle’s acceleration when 𝑠 = 3 m .

  • A βˆ’ 1 9 m/s2
  • B βˆ’ 4 9 m/s2
  • C 4 9 m/s2
  • D βˆ’ 2 2 7 m/s2
  • E βˆ’ 1 6 m/s2

Q13:

A particle started moving along the π‘₯ -axis. When the particle’s displacement from the origin is π‘₯ metres, its velocity is given by 𝑣 = √ 2 ( 3 βˆ’ 3 π‘₯ ) / .  m s Find the particle’s acceleration when its velocity vanished.

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