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Worksheet: Linear Motion with Derivatives

Q1:

A particle is moving in a straight line. After time seconds, where , the body’s displacement relative to a fixed point is given by , where is a fixed unit vector. Find the initial velocity of the particle and its acceleration , 5 seconds after it started moving.

  • A ,
  • B ,
  • C ,
  • D ,

Q2:

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

Q3:

A stone is projected vertically upwards. At time 𝑑 seconds, its height from the ground is given by Determine the speed of the stone when it is 22.5 m high.

Q4:

A particle moves along the π‘₯ -axis. When its displacement from the origin is 𝑠 m, its velocity is given by Find the particle’s minimum velocity.

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

Q5:

A particle moves along the π‘₯ -axis such that at time 𝑑 seconds its velocity is given by After how many seconds is its acceleration equal to 0?

Q6:

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

  • A it is equal to the velocity of the particle
  • B it is equal to π‘₯
  • C it is equal to βˆ’ 𝑣 , where 𝑣 is the velocity of the particle
  • D it is equal to βˆ’ π‘₯

Q7:

A particle started moving along the π‘₯ -axis. At time 𝑑 seconds, its position relative to the origin is given by Find the maximum distance between the particle and the origin π‘₯ m a x , and determine the velocity of the particle 𝑣 when 𝑑 = 3 πœ‹ s .

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

Q8:

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 following equation: Find the maximum speed of the particle 𝑣 m a x and the acceleration of the particle π‘Ž when 𝑣 = 𝑣 m a x .

  • A 𝑣 = 5 . 4 8 / m a x m s , π‘Ž = 0 / m s 2
  • B 𝑣 = 5 . 4 8 / m a x m s , π‘Ž = 1 . 5 / m s 2
  • C 𝑣 = 6 / m a x m s , π‘Ž = 1 . 5 / m s 2
  • D 𝑣 = 6 / m a x m s , π‘Ž = 0 / m s 2

Q9:

A particle is moving in a straight line such that its displacement 𝑠 after 𝑑 seconds is given by Find the velocity of the particle when its acceleration is zero.

Q10:

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 Find the magnitude of its acceleration when its velocity is zero.

  • Aundefined
  • B 10 m/s2
  • C 0 m/s2
  • D 70 m/s2

Q11:

A particle is moving in a straight line such that its velocity 𝑣 at time 𝑑 seconds is given by Find the magnitude of the acceleration of the particle when its velocity is 94 m/s.

Q12:

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

  • A it is equal to the velocity of the particle
  • B it is equal to π‘₯
  • C it is equal to βˆ’ 𝑣 , where 𝑣 is the velocity of the particle
  • D it is equal to βˆ’ π‘₯

Q13:

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

  • A it is equal to the velocity of the particle
  • B it is equal to π‘₯
  • C it is equal to βˆ’ 𝑣 , where 𝑣 is the velocity of the particle
  • D it is equal to βˆ’ π‘₯

Q14:

A particle is moving in a straight line such that its displacement π‘₯ after 𝑑 seconds is given by Determine the time after which the particle changes its direction.

Q15:

A particle is moving in a straight line such that its displacement π‘₯ at time 𝑑 seconds is given by What distance does the particle travel in the first 9.6 seconds?

Q16:

A particle moves along the π‘₯ -axis. At time 𝑑 seconds, its displacement from the origin is given by Determine all the possible values of 𝑑 , in seconds, at which the particle’s speed β€– 𝑣 β€– = 4 m/s.

  • A2, 10
  • B1, 5
  • C 1 5 , 1
  • D 1 2 , 5 2
  • E 2 5 , 2

Q17:

A particle is moving in a straight line such that its displacement from the origin after 𝑑 seconds is given by Find its velocity 𝑣 when 𝑑 = πœ‹ 4 s and its acceleration π‘Ž when 𝑑 = πœ‹ 3 s .

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

Q18:

A particle moves along the π‘₯ -axis. At time 𝑑 seconds, its displacement from the origin is given by When 𝑑 = 1 s , π‘₯ = 7 m , and when 𝑑 = 2 s , the particle’s velocity is 7 m/s. Determine the value of 𝑏 βˆ’ π‘Ž .

Q19:

A particle moves along a straight line. Its displacement at time 𝑑 is π‘₯ = βˆ’ 𝑑 t a n . Find its velocity, 𝑣 , and hence determine which of the following expressions is equal to the acceleration of the particle.

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

Q20:

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

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

Q21:

A particle moves along a straight line. Its displacement at time 𝑑 is π‘₯ = βˆ’ ( 𝑑 ) s i n . 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 π‘₯ .
  • CIt is equal to βˆ’ 𝑣 , where 𝑣 is the velocity of the particle.
  • DIt is equal to βˆ’ π‘₯ .

Q22:

A particle moves along the π‘₯ -axis such that at time 𝑑 seconds its displacement from the origin is given by 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 2
  • B 𝑣 = βˆ’ 3 0 / m s , π‘Ž = 4 0 / m s 2
  • C 𝑣 = βˆ’ 3 0 / m s , π‘Ž = βˆ’ 3 1 . 9 6 / m s 2
  • D 𝑣 = 3 0 / m s , π‘Ž = βˆ’ 4 0 / m s 2
  • E 𝑣 = 3 0 / m s , π‘Ž = 4 0 / m s 2

Q23:

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

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

Q24:

A particle started moving along the -axis. When the particle’s displacement relative to the origin was m in the direction of increasing , its velocity was . Determine the particle’s acceleration when .

Q25:

A particle moves in a straight line such that at time 𝑑 seconds, its velocity is given by After how many seconds does the direction of the particle’s motion change?