Edexcel A-Level Mathematics A2 M2.1.3 Velocity and Acceleration When Questions

Practise differentiating position and velocity functions, locating rest times, finding acceleration, splitting distance at direction changes, and integrating with initial conditions.

Exam points

differentiate x(t) or v(t) to find rest times or acceleration at a given instant
calculate total distance by splitting the interval at changes of direction
integrate v(t) with initial conditions to find displacement or distance from O

[Maximum number: 9]

A particle P moves on the x-axis.

At time t=0, P is instantaneously at rest at O.
At time t seconds, t>0, the x coordinate of P is given by

x=2 t^{\frac{7}{2}}-14 t^{\frac{5}{2}}+\frac{56}{3} t^{\frac{3}{2}}

Find

(a)

the non-zero values of t for which P is at instantaneous rest

[ 3 ]

Use of $v=\frac{dx}{dt}$ .
Recognisable attempt to differentiate the given expression.

$0=7t^{1/2}(t^2-5t+4)$
Set v=0 and solve for t. Dependent on first M1.

$t=1\quad\text{and}\quad t=4$
Correct solution only.

(b)

the total distance travelled by P in the interval 0 <=slant t <=slant 4

[ 3 ]

$s=|x_1-x_0|+|x_4-x_1|$
Correct strategy to find distance for their value(s) of t in [0,4]. Allow M1 if there is no change of direction in the interval.

$=\left|\frac{20}{3}-0\right|+\left|-\frac{128}{3}-\frac{20}{3}\right|$

A1ft

Correct unsimplified expression for their distance, provided there was a change of direction in [0,4].

=56
Correct solution only.

(c)

the acceleration of P when t=4

[ 3 ]

Use of $a=\frac{dv}{dt}$ .
Recognisable attempt to differentiate.

$a=\frac{35}{2}\cdot4^{3/2}-\frac{105}{2}\cdot4^{1/2}+14\cdot4^{-1/2}$
Substitute t=4 in their a and simplify.

=42
Correct solution only.

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Edexcel A-Level Mathematics A2 M2.1.3 Velocity and Acceleration When Questions question 1

Question (a)

Question (b)

Question (c)