CAIE A-Level Mathematics A2 4.2 Kinematics of Motion in a Straight Line Questions

Practise one-dimensional displacement, velocity and acceleration using graphs, calculus and constant-acceleration formulae across single or linked stages of motion.

Exam points

distinguish distance and speed from signed displacement, velocity and acceleration
differentiate or integrate with respect to time and split distance at direction changes
select suvat equations for each constant-acceleration stage with consistent signs

Covered syllabus points

4.2.1—Complex loci 4.2.2—Complex loci 4.2.3—Complex loci 4.2.4—Straight lines

[Maximum number: 5]

Figure for Question 1 — CAIE A-Level Mathematics A2

The displacement of a particle moving in a straight line is s metres at time t seconds after leaving a fixed point O. The particle starts from rest and passes through points P, Q and R, at times t=5, t=10 and t=15 respectively, and returns to O at time t=20. The distances O P, O Q and O R are 50 m , 150 m and 200 m respectively.

The diagram shows a displacement-time graph which models the motion of the particle from t=0 to t=20. The graph consists of two curved segments A B and C D and two straight line segments B C and D E.

(a)

Find the speed of the particle between t=5 and t=10.

[ 1 ]

Speed $=20 \mathrm{~ms}^{-1}$

(b)

Find the acceleration of the particle between t=0 and t=5, given that it is constant.

[ 2 ]

$20=0+a \times 5$
Use of v=u+a t OE

$a=4 \mathrm{~ms}^{-2}$

(c)

Find the average speed of the particle during its motion.

[ 2 ]

$\frac{50+100+50+200}{20}$
Use of $\frac{\text { total distance }}{\text { total time }}$ OE

Average speed $=20 \mathrm{~ms}^{-1}$

[Maximum number: 4]

A car starts from rest and accelerates at $2 \mathrm{~ms}^{-2}$ for 10 s . It then travels at a constant speed for 30 s . The car then uniformly decelerates to rest over a period of 20 s .

(a)

Sketch a velocity-time graph for the motion of the car.

Figure for Question (a) — CAIE A-Level Mathematics A2

[ 2 ]

(b)

Find the total distance travelled by the car.

[ 2 ]

$\begin{aligned} \text { Area }=\frac{1}{2}(30+60) \times 20 \text { or } \text { Area }=\frac{1}{2} \times 10 \times 20+30 \times 20+\frac{1}{2} \times 20 \times 20 \end{aligned}$

M1

Must be considering the area of a trapezium. Allow a single slip in one term only.

Must be adding all terms together if considering two triangles and a rectangle. Using their 20 from (a). If no value for the height shown on the graph in (a), then it must be correct.

$=900 \mathrm{~m}$

A1FT

FT their $20(=45 \times$ their 20) but A0 FT if using 2.

[Maximum number: 7]

A particle travels in a straight line. The velocity of the particle at time $t \mathrm{~s}$ after leaving a point O is $v \mathrm{~m} \mathrm{~s}^{-1}$ , where

v=k t^{2}-4 t+3

The distance travelled by the particle in the first 2 s of its motion is 6 m . You may assume that v>0 in the first 2s of its motion.

(a)

Find the value of k.

[ 4 ]

For attempt at integration

M1*

The power of t must increase by 1 with a change of coefficient in the same term.

Use of s=v t scores M0.

$\frac{1}{2+1} k t^{2+1}-\frac{4}{2} t^{1+1}+3 t\left[=\frac{1}{3} k t^{3}-2 t^{2}+3 t\right][+c]$
Marking guidance:

Allow unsimplified.

$\frac{1}{3} k \times 2^{3}-2 \times 2^{2}+3 \times 2[-0]=6$
Use of limits 0 and 2 with 6 to form an equation in k only
(without c but allow with +c-c ).

k=3

(b)

Find the value of the minimum velocity of the particle. You do not need to show that this velocity is a minimum.

[ 3 ]

$2 \times 3 t-4$

Or at min value $t=\frac{-b}{2 a}=\frac{4}{2 \times 3}$
For attempt at differentiation. Must have expression of the form
a t+b with $a \neq 3$ , unless their $k=\frac{3}{2}$ . Allow 2 k t-4.

$[2 \times 3 t-4=0 \Rightarrow] t=\frac{2}{3}$

A1FT

OE

FT their $k t=\frac{2}{\text { their } k}$ . Allow without working.

$v\left[=3 \times\left(\frac{2}{3}\right)^{2}-4 \times \frac{2}{3}+3\right]=\frac{5}{3} \mathrm{~ms}^{-1}$
OE

Marking guidance:

Allow 1.67 or better for v.

Alternative Method for Question 4(b): Using completing the square

Attempt at completing the square
Must have $\left(t-\frac{2}{3}\right)^{2}$ OE, or $\left(t-\frac{2}{\text { their } k}\right)^{2}$ .

$3\left(t-\frac{2}{3}\right)^{2}-\frac{4}{3}+3$

(A1FT)

FT their $k k\left(t-\frac{2}{k}\right)^{2}-\frac{4}{k}+3$ .

$v=\frac{5}{3} \mathrm{~m} \mathrm{~s}^{-1}$
OE

Allow 1.67 or better.

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question (a)

Question (b)

Question 3

Question (a)

Question (b)