Edexcel IAL Mathematics M2.1.3 velocity and acceleration functions

Practise using displacement, velocity and acceleration functions of time to find rest times, acceleration values and distances travelled.

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 $v=\frac{dx}{dt}$ .

M1

Recognisable attempt to differentiate the given expression.

$0=7t^{1/2}(t^2-5t+4)$

DM1

Set v=0 and solve for t. Dependent on first M1.

$t=1\quad\text{and}\quad t=4$

A1

Correct solution only.

(b)

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

[ 3 ]

$s=|x_1-x_0|+|x_4-x_1|$

M1

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

A1

Correct solution only.

(c)

the acceleration of P when t=4

[ 3 ]

Use of $a=\frac{dv}{dt}$ .

M1

Recognisable attempt to differentiate.

$a=\frac{35}{2}\cdot4^{3/2}-\frac{105}{2}\cdot4^{1/2}+14\cdot4^{-1/2}$

M1

Substitute t=4 in their a and simplify.

=42

A1

Correct solution only.

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M2.1.3 - Velocity and acceleration when question 1

Question (a)

Question (b)

Question (c)