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€Ή3π‘‘βˆ’54𝑑+38𝑑𝑑β‰₯0.m, Determine the time interval during which the velocity of the particle is increasing.

  • A(12,∞)
  • B(18,∞)
  • C(36,∞)
  • 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=𝑑+3𝑑+7. Find the particle’s average velocity between 𝑑=2s and 𝑑=4s.

  • A18 m/s
  • B11 m/s
  • Cβˆ’2 m/s
  • D9 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€Ήπ‘‘βˆ’3𝑑𝑑β‰₯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€Ήπ‘‘βˆ’4𝑑𝑑β‰₯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 meters is given as a function of time 𝑑 in seconds by 𝑠=5π‘‘βˆ’84𝑑+33𝑑𝑑β‰₯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€Ήβˆ’π‘‘+4𝑑β‰₯0.m, Determine the time at which the particle’s acceleration is 9 m/s2.

  • A157 s
  • B412 s
  • C112 s
  • D212 s
  • E3 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€Ήπ‘‘βˆ’π‘‘βˆ’3,𝑑β‰₯0.m Determine whether the particle is accelerating or decelerating when 𝑑=2s.

  • Adecelerating
  • Baccelerating

Q8:

A body moves along the π‘₯-axis such that at time 𝑑 seconds, its displacement from the origin is given by 𝑠=ο€Ή6π‘‘βˆ’π‘‘βˆ’3π‘‘βˆ’4𝑑+3𝑑β‰₯0.οŠͺm, What is its velocity when its acceleration is equal to 0?

  • Aβˆ’138 m/s
  • Bβˆ’139 m/s
  • Cβˆ’6913 m/s
  • Dβˆ’499 m/s
  • Eβˆ’449 m/s

Q9:

A particle is moving in a straight line such that its displacement 𝑠 at 𝑑 seconds is given by 𝑠=ο€Ή5π‘‘βˆ’30𝑑+4𝑑𝑑β‰₯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’12𝑑+12π‘‘βˆ’3π‘‘οŠ©οŠ¨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
  • B12, 10
  • C27, 25
  • Dβˆ’12, 12

Q11:

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

  • Aο€Ό0,32
  • B(0,3]
  • C(3,∞)
  • Dο€Όβˆ’βˆž,32
  • Eο€Ό32,3

Q12:

A particle moves along the π‘₯-axis. When its displacement from the origin is 𝑠 m, its velocity is given by 𝑣=43+𝑠/.ms Find the particle’s acceleration when 𝑠=3m.

  • A49 m/s2
  • Bβˆ’19 m/s2
  • Cβˆ’16 m/s2
  • Dβˆ’227 m/s2
  • Eβˆ’49 m/s2

Q13:

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

Q14:

A particle moves along the π‘₯-axis. Its velocity, 𝑣, in meters per second is given by 𝑣=9(π‘‘βˆ’4)βˆ’12(π‘‘βˆ’4), where 𝑑 is the time in seconds. Find the interval over which the particle is accelerating in the positive π‘₯-direction.

  • Aο€Ό72,92
  • Bο€Ό154,174
  • Cο€Ό0,92
  • Dο€Ό4,92
  • E72,92

Q15:

A particle moves along the π‘₯-axis. Its displacement from the origin is 𝑠 meters at a time 𝑑 seconds. The displacement is given by the equation 𝑠=2ο€Ό5πœ‹7π‘‘οˆ+7sin. Find the times 𝑑 at which the particle’s velocity is equal to 0.

  • A𝑑=57𝑛, where 𝑛 is an integer
  • B𝑑=75+145𝑛, where 𝑛 is an integer
  • C𝑑=710+75𝑛, where 𝑛 is an integer
  • D𝑑=514+57𝑛, where 𝑛 is an integer
  • E𝑑=75𝑛, where 𝑛 is an integer

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