Worksheet: Linear Motion with Derivatives

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. When the particle’s displacement relative to the origin was s m in the direction of increasing π‘₯, its velocity was vs=5/ms. Determine the particle’s acceleration when s=13m.

Q2:

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.

  • A π‘₯ = 1 5 m a x m , 𝑣 = βˆ’ 1 6 8 / m s
  • B π‘₯ = 7 2 5 m a x m , 𝑣 = βˆ’ 1 6 8 / m s
  • C π‘₯ = 1 5 m a x m , 𝑣 = 1 6 8 / m s
  • D π‘₯ = 7 2 5 m a x m , 𝑣 = 1 6 8 / m s

Q3:

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.

Q4:

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?

Q5:

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.

Q6:

A particle is moving in a straight line such that its velocity 𝑣 and position π‘₯ satisfy the following equation: 𝑣=16(54βˆ’π‘₯). Find an expression for the particle’s acceleration in terms of π‘₯.

  • A π‘₯ 3 ( π‘₯ βˆ’ 5 4 )  
  • B π‘₯ 6 ( π‘₯ βˆ’ 5 4 )  
  • C βˆ’ π‘₯ 2 π‘₯ βˆ’ 5 4
  • D βˆ’ 1 1 2 π‘₯

Q7:

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.

Q8:

A particle is moving in a straight line such that its displacement π‘₯ at time 𝑑 seconds is given by π‘₯=ο€Ή9.6π‘‘βˆ’π‘‘ο…π‘‘β‰₯0.m, What distance does the particle travel in the first 9.6 seconds?

Q9:

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.

Q10:

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.

  • A 2 𝑣 π‘₯
  • B 𝑣 π‘₯
  • C βˆ’ 𝑣 π‘₯
  • D βˆ’ 2 𝑣 π‘₯

Q11:

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
  • B 2 5 , 2
  • C1, 5
  • D 1 5 , 1
  • E 1 2 , 5 2

Q12:

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 √ 7 4 9 m/s
  • B √ 7 7 m/s
  • C √ 2 1 4 m/s
  • D √ 2 2 8 m/s
  • E √ 7 9 8 m/s

Q13:

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.

  • A 𝑣 = βˆ’ 3 2 . 3 2 / m s , π‘Ž = βˆ’ 3 1 . 9 6 / m s 
  • B 𝑣 = 3 0 / m s , π‘Ž = βˆ’ 4 0 / m s 
  • C 𝑣 = 3 0 / m s , π‘Ž = 4 0 / m s 
  • D 𝑣 = βˆ’ 3 0 / m s , π‘Ž = βˆ’ 3 1 . 9 6 / m s 
  • E 𝑣 = βˆ’ 3 0 / m s , π‘Ž = 4 0 / m s 

Q14:

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, 5 seconds after it started moving.

  • A v c   = ο€Ό ο€Ό 5 2 𝑑 + 5   / m s , a c = ( 2 5 ) / m s 
  • B v c  = ( 5 ) / m s , a c = ( 3 0 ) / m s 
  • C v c   = ο€Ό ο€Ό 5 2 𝑑 + 5   / m s , a c = ( 3 0 ) / m s 
  • D v c  = ( 5 ) / m s , a c = ( 2 5 ) / m s 

Q15:

A particle moves along the π‘₯-axis such that at time 𝑑 seconds its velocity is given by 𝑣=ο€Ήπ‘‘βˆ’12𝑑+3/𝑑β‰₯0.ms, After how many seconds is its acceleration equal to 0?

Q16:

A particle moves along a straight line. Its displacement at time 𝑑 is π‘₯=βˆ’(𝑑)cos. Which of the following statements about the acceleration of the particle is true?

  • Ait is equal to π‘₯
  • Bit is equal to βˆ’π‘£, where 𝑣 is the velocity of the particle
  • Cit is equal to the velocity of the particle
  • Dit is equal to βˆ’π‘₯

Q17:

A particle is moving in a straight line such that its speed 𝑣, measured in meters per second, and its position π‘₯, measured in meters, satisfy the equation 𝑣=33βˆ’3π‘₯cos. Find the maximum speed of the particle 𝑣max and the acceleration of the particle π‘Ž when 𝑣=𝑣max.

  • A 𝑣 = 5 . 4 8 / m a x m s , π‘Ž = 0 / m s 
  • B 𝑣 = 6 / m a x m s , π‘Ž = 1 . 5 / m s 
  • C 𝑣 = 5 . 4 8 / m a x m s , π‘Ž = 1 . 5 / m s 
  • D 𝑣 = 6 / m a x m s , π‘Ž = 0 / m s 

Q18:

A particle is moving in a straight line. The relation between its velocity 𝑣, measured in meters per second, and its position π‘₯, measured in meters, is given by 𝑣=7ο€Ή100βˆ’π‘₯ο…οŠ¨οŠ¨. Find the magnitude of its acceleration when its velocity is zero.

Q19:

A particle is moving in a straight line such that its velocity 𝑣 at time 𝑑 seconds is given by 𝑣=ο€Ή2π‘‘βˆ’68/𝑑β‰₯0.ms, Find the magnitude of the acceleration of the particle when its velocity is 94 m/s.

Q20:

A particle moves along a straight line. Its displacement at time 𝑑 is π‘₯=(𝑑)cos. Which of the following statements about the acceleration of the particle is true?

  • Ait is equal to βˆ’π‘₯
  • Bit is equal to π‘₯
  • Cit is equal to βˆ’π‘£ , where 𝑣 is the velocity of the particle
  • Dit is equal to the velocity of the particle

Q21:

A particle moves along a straight line. Its displacement at time 𝑑 is π‘₯=𝑑sin. Which of the following statements about the acceleration of the particle is true?

  • Ait is equal to βˆ’π‘₯
  • Bit is equal to the velocity of the particle
  • Cit is equal to βˆ’π‘£, where 𝑣 is the velocity of the particle
  • Dit is equal to π‘₯

Q22:

A particle is moving in a straight line such that its displacement from the origin after 𝑑 seconds is given by π‘₯=ο€Ό132π‘‘οˆπ‘‘β‰₯0.cosm, Find its velocity 𝑣 when 𝑑=πœ‹4s and its acceleration π‘Ž when 𝑑=πœ‹3s.

  • A 𝑣 = βˆ’ 2 3 / m s , π‘Ž = 2 3 / m s 
  • B 𝑣 = βˆ’ 2 3 / m s , π‘Ž = βˆ’ 2 3 / m s 
  • C 𝑣 = βˆ’ 4 3 / m s , π‘Ž = 4 3 / m s 
  • D 𝑣 = 4 3 / m s , π‘Ž = βˆ’ 4 3 / m s 

Q23:

A particle moves along the π‘₯-axis. At time 𝑑 seconds, its displacement from the origin is given by π‘₯=ο€Ήπ‘Žπ‘‘βˆ’π‘‘+𝑏𝑑β‰₯0.m, When 𝑑=1s, π‘₯=7m, and when 𝑑=2s, the particle’s velocity is 7 m/s. Determine the value of π‘βˆ’π‘Ž.

Q24:

A particle moves along a straight line. Its displacement at time 𝑑 is π‘₯=βˆ’π‘‘tan. Find its velocity, 𝑣, and hence determine which of the following expressions is equal to the acceleration of the particle.

  • A 𝑣 π‘₯
  • B βˆ’ 2 𝑣 π‘₯
  • C 2 𝑣 π‘₯
  • D βˆ’ 𝑣 π‘₯

Q25:

A particle moves along a straight line. Its displacement at time 𝑑 is π‘₯=βˆ’(𝑑)sin. Which of the following statements about the acceleration of the particle is true?

  • AIt is equal to the velocity of the particle.
  • BIt is equal to βˆ’π‘₯.
  • CIt is equal to π‘₯.
  • DIt is equal to βˆ’π‘£, where 𝑣 is the velocity of the particle.

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