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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 , a particle leaves the origin and moves along the . At time s, the velocity of in metres per second is given by . What is the maximum velocity of ?

**Q2: **

A particle P moves in a straight line. Its displacement, m, from a fixed point at time s is given by , where . The diagram shows a displacementβtime graph for the motion of P.

At what time is P moving with minimum velocity?

- A1 s
- B2 s
- C0 s
- D s
- E s

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

**Q3: **

A particle P passes through a point and moves in a straight line. Its displacement, m, from at time is given by . Calculate the values of at which P is instantaneously at rest.

- A,
- B,
- C,
- D,
- E,

Find the value of at which the acceleration is zero.

- A s
- B13 s
- C26 s
- D s
- E s

What is the maximum velocity of the particle?

- A169 m/s
- B m/s
- C m/s
- D m/s
- E121 m/s

**Q4: **

A particle starts at a fixed point and moves in a straight line. Its distance, m, from at time is given by , where . Find the greatest distance of from , giving your answer to 1 decimal place.

**Q5: **

A particle P moves along the . Its velocity, m/s in the positive -direction, at time s is given by , for . 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
- CProving that the discriminant of the velocity function is negative
- DProving that the acceleration function has no zeros for
- 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, m, above ground level at time s is given by the equation , for . The diagram below shows a sketch of the displacementβtime graph for the helicopter.

Work out the maximum and minimum heights of the helicopter.

- AMaximum height , minimum height
- BMaximum height , minimum height
- CMaximum height , minimum height
- DMaximum height , minimum height
- EMaximum height , minimum height

**Q7: **

A particle moves along the so that its velocity, m/s, at time s is given by , where . Find the value of when the particle reaches its maximum velocity.

Find the acceleration when the velocity is zero.