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 π‘₯=5𝑑+4π‘‘οŠ¨ and 𝑦=3π‘‘βˆ’2. Find the speed of the particle at 𝑑=2.

  • A27
  • B9√7
  • C3√65
  • D585
  • E567

Q2:

A particle has a position defined by the equations π‘₯=π‘‘βˆ’5π‘‘οŠ© and 𝑦=3βˆ’2π‘‘οŠ¨. Find the velocity vector of the particle at 𝑑=2.

  • Avij=42βˆ’12
  • Bvij=8βˆ’7
  • Cvij=12βˆ’4
  • Dvij=7βˆ’8
  • Evij=7+8

Q3:

A moving particle along a curve is defined by the two equations π‘₯=π‘‘βˆ’3𝑑+4 and 𝑦=7π‘‘βˆ’3. Find the acceleration vector of the particle at 𝑑=1.

  • Aaij=14+16
  • Baij=βˆ’16+14
  • Caij=βˆ’7+14
  • Daij=7+14
  • Eaij=16+14

Q4:

A moving particle is defined by the two equations π‘₯=π‘‘βˆ’5π‘‘βˆ’5 and 𝑦=7π‘‘βˆ’3. Find the magnitude of the acceleration of the particle at 𝑑=1.

  • A2√58
  • B232
  • C4√10
  • D10√2
  • E200

Q5:

A moving particle is defined by the two equations ddπ‘₯𝑑=3π‘‘βˆ’2 and ddοŠ¨οŠ¨π‘¦π‘‘=2𝑑+6. Find the magnitude of the acceleration of the particle at 𝑑=3.

  • A√193
  • B193
  • C√5
  • D√13
  • E13

Q6:

Given that the position of a moving particle is defined by the parametric equations π‘₯=5𝑑𝑑+2 and 𝑦=6π‘‘π‘‘βˆ’3, find the speed of the particle at 𝑑=1 to the nearest tenth.

  • A𝑣=34.8
  • B𝑣=2.7
  • C𝑣=37.2
  • D𝑣=5.9
  • E𝑣=6.1

Q7:

If a particle is moving on a curve defined by the parametric equations π‘₯=12π‘‘βˆ’4𝑑+3 and 𝑦=12𝑑+3π‘‘οŠ¨, find the time, to the nearest tenth, at which 𝑣=64.

  • A𝑑=45.6
  • B𝑑=0.9
  • C𝑑=44.6
  • D𝑑=3.9
  • E𝑑=4.9

Q8:

If a particle is moving on a curve defined by the parametric equations ddπ‘₯𝑑=π‘‘βˆ’6π‘‘οŠ¨ and dd𝑦𝑑=𝑑+5π‘‘βˆ’3, find the time, to the nearest tenth, at which π‘Ž=64.

  • A𝑑=0.9
  • B𝑑=22.7
  • C𝑑=6.0
  • D𝑑=22.3
  • E𝑑=0.4

Q9:

Suppose that a particle is moving on a curve defined by the parametric equations ddπ‘₯𝑑=5π‘‘βˆ’15 and dd𝑦𝑑=8βˆ’4𝑑. If the particle is initially at horizontal displacement 𝑑=32.3, find the minimum horizontal displacement from 𝑑=0.

  • A𝑑=9.8
  • B𝑑=32.3
  • C𝑑=38.3
  • D𝑑=6
  • E𝑑=22.5

Q10:

Suppose that a particle is moving on a curve defined by the parametric equations ddπ‘₯𝑑=π‘‘βˆ’3 and dd𝑦𝑑=12βˆ’6𝑑. If the particle is initially at vertical displacement 𝑑=9.5, find the maximum positive vertical displacement from 𝑑=0.

  • A𝑑=12
  • B𝑑=9.5
  • C𝑑=5
  • D𝑑=14.75
  • E𝑑=2

Q11:

A particle starts its motion at the point 2+4ij. The particle’s velocity is given by the parametric equations ddπ‘₯𝑑=4𝑑+3π‘‘οŠ¨ and dd𝑦𝑑=6𝑑+2, where 𝑑 is the time after the particle started its motion. Find the position vector of the particle at time 𝑑.

  • Aο€Ή2𝑑+3𝑑+2+ο€Ή2𝑑+2𝑑+4ο…οŠ©οŠ¨οŠͺij
  • Bο€Ό43𝑑+32𝑑+2+ο€Ό32𝑑+2𝑑+4οŠͺij
  • Cο€Ή4𝑑+3π‘‘βˆ’2+ο€Ή6𝑑+2π‘‘βˆ’4ο…οŠ©οŠ¨οŠͺij
  • Dο€Ό43𝑑+32π‘‘οˆ+ο€Ό32𝑑+2π‘‘οˆοŠ©οŠ¨οŠͺij
  • Eο€Ή4𝑑+3𝑑+2+ο€Ή6𝑑+6ο…οŠ¨οŠ©ij

Q12:

The position of a particle is defined by π‘₯=1𝑑+1 and 𝑦=3𝑑+7π‘‘οŠ¨. Find the speed of the particle when 𝑑=3, giving your answer to two decimal places.

Q13:

A particle moves along a curve defined by the parametric equations π‘₯=2 and 𝑦=8. What is the shape of the trajectory of the particle?

  • AA parabola
  • BA cubic curve
  • CAn exponential curve
  • DA hyperbola
  • EA logarithmic curve

Q14:

The acceleration of a particle in a coordinate grid is given by the parametric equations ddcosπ‘₯𝑑=4𝑑 and ddοŠ¨οŠ¨π‘¦π‘‘=π‘’ο‘‰οŽ‘. At the time 𝑑=0, the position of the particle is given by π‘₯=βˆ’3 and 𝑦=9 and the velocity of the particle is given by ddπ‘₯𝑑=4 and dd𝑦𝑑=βˆ’1. Find the parametric equations which represent the position of the particle at time 𝑑.

  • Aπ‘₯=βˆ’16(4𝑑)𝑦=14𝑒cosandο‘‰οŽ‘
  • Bπ‘₯=βˆ’116(4𝑑)+4π‘‘βˆ’4716𝑦=4π‘’βˆ’3𝑑+5cosandο‘‰οŽ‘
  • Cπ‘₯=βˆ’116(4𝑑)+4π‘‘βˆ’3𝑦=4π‘’βˆ’π‘‘+9cosandο‘‰οŽ‘
  • Dπ‘₯=14(4𝑑)+4𝑦=2π‘’βˆ’3sinandο‘‰οŽ‘
  • Eπ‘₯=βˆ’(4𝑑)+4π‘‘βˆ’4𝑦=π‘’βˆ’2𝑑+8cosandο‘‰οŽ‘

Q15:

A particle’s velocity is given by the parametric equations ddπ‘₯𝑑=2𝑑+5π‘‘βˆ’3𝑑+1 and dd𝑦𝑑=βˆšπ‘‘+3. Find the acceleration vector of the particle at 𝑑=2.

  • A42βˆ’6√1111ij
  • B42+6√1111ij
  • C15βˆ’6√1111ij
  • D31+√11ij
  • E41+6√1111ij

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.