Worksheet: Planar Motion Using Parametric Equations

In this worksheet, we will practice describing motion of a particle along a curve defined by parametric functions.

Q1:

A particle has a position defined by the equations 𝑥 = 5 𝑡 + 4 𝑡 and 𝑦 = 3 𝑡 2 . Find the speed of the particle at 𝑡 = 2 .

  • A567
  • B 3 6 5
  • C27
  • D585
  • E 9 7

Q2:

A particle has a position defined by the equations 𝑥 = 𝑡 5 𝑡 and 𝑦 = 3 2 𝑡 . Find the velocity vector of the particle at 𝑡 = 2 .

  • A v i j = 1 2 4
  • B v i j = 4 2 1 2
  • C v i j = 8 7
  • D v i j = 7 + 8
  • E v i j = 7 8

Q3:

A moving particle along a curve is defined by the two equations 𝑥 = 𝑡 3 𝑡 + 4 and 𝑦 = 7 𝑡 3 . Find the acceleration vector of the particle at 𝑡 = 1 .

  • A a i j = 1 6 + 1 4
  • B a i j = 7 + 1 4
  • C a i j = 1 6 + 1 4
  • D a i j = 1 4 + 1 6
  • E a i j = 7 + 1 4

Q4:

A moving particle is defined by the two equations 𝑥 = 𝑡 5 𝑡 5 and 𝑦 = 7 𝑡 3 . Find the magnitude of the acceleration of the particle at 𝑡 = 1 .

  • A 4 1 0
  • B 1 0 2
  • C200
  • D232
  • E 2 5 8

Q5:

A moving particle is defined by the two equations d d 𝑥 𝑡 = 3 𝑡 2 and d d 𝑦 𝑡 = 2 𝑡 + 6 . Find the magnitude of the acceleration of the particle at 𝑡 = 3 .

  • A193
  • B 1 3
  • C 1 9 3
  • D 5
  • E13

Q6:

Given that the position of a moving particle is defined by the parametric equations 𝑥 = 5 𝑡 𝑡 + 2 and 𝑦 = 6 𝑡 𝑡 3 , find the speed of the particle at 𝑡 = 1 to the nearest tenth.

  • A 𝑣 = 2 . 7
  • B 𝑣 = 3 7 . 2
  • C 𝑣 = 6 . 1
  • D 𝑣 = 5 . 9
  • E 𝑣 = 3 4 . 8

Q7:

If a particle is moving on a curve defined by the parametric equations 𝑥 = 1 2 𝑡 4 𝑡 + 3 and 𝑦 = 1 2 𝑡 + 3 𝑡 , find the time, to the nearest tenth, at which 𝑣 = 6 4 .

  • A 𝑡 = 3 . 9
  • B 𝑡 = 0 . 9
  • C 𝑡 = 4 . 9
  • D 𝑡 = 4 4 . 6
  • E 𝑡 = 4 5 . 6

Q8:

If a particle is moving on a curve defined by the parametric equations d d 𝑥 𝑡 = 𝑡 6 𝑡 and d d 𝑦 𝑡 = 𝑡 + 5 𝑡 3 , find the time, to the nearest tenth, at which 𝑎 = 6 4 .

  • A 𝑡 = 6 . 0
  • B 𝑡 = 2 2 . 7
  • C 𝑡 = 0 . 9
  • D 𝑡 = 2 2 . 3
  • E 𝑡 = 0 . 4

Q9:

Suppose that a particle is moving on a curve defined by the parametric equations d d 𝑥 𝑡 = 5 𝑡 1 5 and d d 𝑦 𝑡 = 8 4 𝑡 . If the particle is initially at horizontal displacement 𝑑 = 3 2 . 3 , find the minimum horizontal displacement from 𝑑 = 0 .

  • A 𝑑 = 3 2 . 3
  • B 𝑑 = 3 8 . 3
  • C 𝑑 = 9 . 8
  • D 𝑑 = 2 2 . 5
  • E 𝑑 = 6

Q10:

Suppose that a particle is moving on a curve defined by the parametric equations d d 𝑥 𝑡 = 𝑡 3 and d d 𝑦 𝑡 = 1 2 6 𝑡 . If the particle is initially at vertical displacement 𝑑 = 9 . 5 , find the maximum positive vertical displacement from 𝑑 = 0 .

  • A 𝑑 = 1 2
  • B 𝑑 = 5
  • C 𝑑 = 9 . 5
  • D 𝑑 = 1 4 . 7 5
  • E 𝑑 = 2

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