# 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 displacement–time 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 displacement–time 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.

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