Lesson Worksheet: Applications of Derivatives on Rectilinear Motion Mathematics • Higher Education

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 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.

Q2:

A particle moves in a straight line such that at time 𝑑 seconds its position 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 in the direction of motion when 𝑑=2s.

  • ADecelerating
  • BAccelerating

Q3:

A particle is moving in a straight line along the π‘₯-axis such that its displacement 𝑠 meters after 𝑑 seconds is given by 𝑠(𝑑)=βˆ’π‘‘+6𝑑+2π‘‘οŠ©οŠ¨. Find the maximum velocity of the particle in the positive π‘₯-direction.

Q4:

A particle moves along the π‘₯-axis. Its position from the origin is 𝑠 meters at a time 𝑑 seconds. The position 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

Q5:

A particle is moving in a straight line such that its displacement 𝑆 meters after 𝑑 seconds is given by 𝑆=4π‘‘βˆ’55𝑑+208π‘‘οŠ©οŠ¨.

Find the velocity of the particle at 𝑑=8seconds.

Find the time interval during which the velocity of the particle is decreasing.

  • A0≀𝑑<554
  • B0≀𝑑<556
  • C0≀𝑑<5512
  • D0≀𝑑<559
  • E0≀𝑑<5524

Q6:

A particle is moving in a straight line such that its displacement 𝑠 at time 𝑑 is given by 𝑠=13π‘‘βˆ’14π‘‘οŠ©οŠ¨m.

What is the total distance covered by the particle during the first 2 seconds? Write your answer approximate to two decimal places.

Q7:

A particle moves in a straight line, with respect to a stationary point, with position vector ri=ο€Ήβˆ’2𝑑+5𝑑+3ο…οŠ¨, where 𝑑β‰₯0 and is measured in seconds. i is a unit vector parallel to the straight line, and r is measured in meters.

Find the magnitude of the displacement vector after 2 s.

Find the total distance covered by the particle after 2 s.

Determine the magnitude of the average velocity vector of the particle between 𝑑=0s and 𝑑=2s.

Determine the average speed of the particle between 𝑑=0s and 𝑑=2s.

Q8:

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

Q9:

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.

Q10:

A particle moves along the 𝑋-axis. When its displacement from the origin is 𝑠 m, its velocity is given by 𝑣=ο„ž1βˆ’14𝑠+98/.ms Find the particle’s minimum velocity.

  • A√749 m/s
  • B√77 m/s
  • C√214 m/s
  • D√228 m/s
  • E√798 m/s

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