Lesson Worksheet: Linear Motion with Derivatives Mathematics

In this worksheet, we will practice using differentiation to find instantaneous velocity, speed, and acceleration of a particle.

Q1:

A particle started moving along the π‘₯-axis. At time 𝑑 seconds, its position relative to the origin is given by π‘₯=[12(14𝑑)+9(14𝑑)],𝑑β‰₯0.sincosm Find the maximum distance between the particle and the origin π‘₯max, and determine the velocity of the particle 𝑣 when 𝑑=3πœ‹s.

  • Aπ‘₯=15maxm, 𝑣=βˆ’168/ms
  • Bπ‘₯=725maxm, 𝑣=βˆ’168/ms
  • Cπ‘₯=15maxm, 𝑣=168/ms
  • Dπ‘₯=725maxm, 𝑣=168/ms

Q2:

A particle moves along the π‘₯-axis such that at time 𝑑 seconds its displacement from the origin is given by π‘₯=(102𝑑)𝑑β‰₯0.sinm, Determine the particle’s acceleration when π‘₯=βˆ’5m.

Q3:

A particle moves along the π‘₯-axis so that its position relative to the origin after time 𝑑 (where 𝑑β‰₯0) is given by π‘Ÿ=5𝑑+12𝑑.sincos What is the particle’s maximum displacement?

Q4:

A stone is projected vertically upward. At time 𝑑 seconds, its height from the ground is given by β„Ž=ο€Ή46.6π‘‘βˆ’4.9𝑑𝑑β‰₯0.m, Determine the speed of the stone when it is 22.5 m high.

Q5:

A particle is moving in a straight line such that its displacement 𝑠 after 𝑑 seconds is given by 𝑠=ο€Ήβˆ’10𝑑+12𝑑+10𝑑𝑑β‰₯0.m, Find the velocity of the particle when its acceleration is zero.

Q6:

A particle is moving in a straight line such that its displacement π‘₯ after 𝑑 seconds is given by π‘₯=ο€Ή2π‘‘βˆ’24π‘‘βˆ’15𝑑β‰₯0.m, Determine the time after which the particle changes its direction.

Q7:

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

  • A2𝑣π‘₯
  • B𝑣π‘₯
  • Cβˆ’π‘£π‘₯
  • Dβˆ’2𝑣π‘₯

Q8:

A particle moves along the π‘₯-axis. At time 𝑑 seconds, its displacement from the origin is given by π‘₯=ο€Ή2π‘‘βˆ’6π‘‘βˆ’4𝑑β‰₯0.m, Determine all the possible values of 𝑑, in seconds, at which the particle’s speed |𝑣|=4 m/s.

  • A2, 10
  • B25, 2
  • C1, 5
  • D15, 1
  • E12, 52

Q9:

A particle moves along the π‘₯-axis such that at time 𝑑 seconds its displacement from the origin is given by π‘₯=[15(2𝑑)+10(2𝑑)+118],𝑑β‰₯0.sincosm Find the particle’s velocity, 𝑣, and acceleration, π‘Ž, at 𝑑=πœ‹s.

  • A𝑣=βˆ’32.32/ms, π‘Ž=βˆ’31.96/ms
  • B𝑣=30/ms, π‘Ž=βˆ’40/ms
  • C𝑣=30/ms, π‘Ž=40/ms
  • D𝑣=βˆ’30/ms, π‘Ž=βˆ’31.96/ms
  • E𝑣=βˆ’30/ms, π‘Ž=40/ms

Q10:

A particle is moving in a straight line. After time 𝑑 seconds, where 𝑑β‰₯0, the body’s displacement relative to a fixed point is given by sc=56𝑑+5π‘‘οˆο οŠ©m, where c is a fixed unit vector. Find the initial velocity of the particle v and its acceleration a, Β 5 seconds after it started moving.

  • Avc=ο€Όο€Ό52𝑑+5/ms, ac=(25)/ms
  • Bvc=(5)/ms, ac=(30)/ms
  • Cvc=ο€Όο€Ό52𝑑+5/ms, ac=(30)/ms
  • Dvc=(5)/ms, ac=(25)/ms

This lesson includes 17 additional questions and 171 additional question variations for subscribers.

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