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Lesson Worksheet: Maxima and Minima in Linear Motion Mathematics

In this worksheet, we will practice finding the maximum and minimum values for the displacement and velocity of a particle moving in a straight line.

Q1:

At time 𝑑=0, a particle P leaves the origin and moves along the π‘₯-axis. At time 𝑑 s, the velocity of P in metres per second is given by 𝑣=10π‘‘βˆ’π‘‘/ms. What is the maximum velocity of P?

Q2:

A particle P moves in a straight line. Its displacement, 𝑠 m, from a fixed point 𝑂 at time 𝑑 s is given by 𝑠=π‘‘βˆ’π‘‘βˆ’π‘‘+3, where 0≀𝑑≀2. The diagram shows a displacement–time graph for the motion of P.

At what time is P moving with minimum velocity?

  • A1 s
  • B2 s
  • C0 s
  • D16 s
  • E13 s

Find the displacement of P from 𝑂 at this time, giving your answer to 2 decimal places.

Q3:

A particle P passes through a point 𝑂 and moves in a straight line. Its displacement, 𝑠 m, from 𝑂 at time 𝑑 is given by 𝑠=βˆ’π‘‘+13π‘‘βˆ’16π‘‘οŠ©οŠ¨. Calculate the values of 𝑑 at which P is instantaneously at rest.

  • A𝑑=43s, 𝑑=4s
  • B𝑑=23s, 𝑑=4s
  • C𝑑=43s, 𝑑=8s
  • D𝑑=4s, 𝑑=8s
  • E𝑑=23s, 𝑑=8s

Find the value of 𝑑 at which the acceleration is zero.

  • A136 s
  • B13 s
  • C26 s
  • D263 s
  • E133 s

What is the maximum velocity of the particle?

  • A169 m/s
  • B252227 m/s
  • C1213 m/s
  • D1693 m/s
  • E121 m/s

Q4:

A particle P starts at a fixed point 𝑂 and moves in a straight line. Its distance, 𝑠 m, from 𝑂 at time 𝑑 is given by 𝑠=π‘‘βˆ’14𝑑+49𝑑οŠͺ, where 0≀𝑑≀7. Find the greatest distance of P from 𝑂, giving your answer to 1 decimal place.

Q5:

A particle P moves along the π‘₯-axis. Its velocity, 𝑣 m/s in the positive π‘₯-direction, at time 𝑑 s is given by 𝑣=2π‘‘βˆ’2𝑑+3, for 𝑑β‰₯0. What method can be used to show that the particle never comes to rest?

  • AProving that the discriminant of the velocity function is positive
  • BProving that the acceleration function is positive for 𝑑β‰₯0
  • CProving that the discriminant of the velocity function is negative
  • DProving that the acceleration function has no zeros for 𝑑β‰₯0
  • EProving that the acceleration function has no zeros

Find the minimum velocity of P.

Q6:

A remote-controlled helicopter hovers in the air. Its vertical height, 𝑠 m, above ground level at time 𝑑 s is given by the equation 𝑠=150ο€Ήπ‘‘βˆ’12𝑑+16𝑑+1100οŠͺ, for 0<𝑑<10. The diagram below shows a sketch of the displacement–time graph for the helicopter.

Work out the maximum and minimum heights of the helicopter.

  • AMaximum height =22m, minimum height =1.52m
  • BMaximum height =22.1m, minimum height =1.52m
  • CMaximum height =76m, minimum height =1105m
  • DMaximum height =1.52m, minimum height =22.1m
  • EMaximum height =1105m, minimum height =76m

Q7:

A particle moves along the π‘₯-axis so that its velocity, 𝑣 m/s, at time 𝑑 s is given by 𝑣=βˆ’5𝑑+20𝑑+60, where 0≀𝑑≀6. Find the value of 𝑑 when the particle reaches its maximum velocity.

Find the acceleration when the velocity is zero.

This lesson includes 63 additional question variations for subscribers.

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