Edexcel A-Level Mathematics A2 M2.1 Kinematics of a Particle Moving in a Straight Line or Plane Questions

Practise kinematics for particles in lines and planes, linking suvat, projectiles, vectors, graphs, differentiation and integration with motion.

Exam points

use suvat or projectile components to find times, speeds and distances
differentiate displacement or velocity functions to find rest times and acceleration
interpret speed-time graphs and vector directions from calculated motion values

Covered syllabus points

M2.1.1 - Motion in a vertical plane with constant acceleration, e M2.1.2 - Simple cases of motion of a projectile M2.1.3 - Velocity and acceleration when M2.1.4 - Differentiating and integrating vectors

[Maximum number: 5]

Find the exact value of k

Hemisphere / Cone

$\begin{aligned} \frac{2}{3}\times\frac{3}{8}r &= \frac{k}{3}\times\frac{1}{4}kr,\\ k^2&=3,\quad k=\sqrt{3}. \end{aligned}$
M1: Form a dimensionally correct moments equation with the correct value for $\bar{X}$ .
A1ft: Correct equation, follow through their masses and distances;
signs correct.

\begin{array}{lll} \text{Mass} & \frac{2}{3}\pi r^3 & \frac{1}{3}\pi k r^3\\ \text{Distance of centre of mass from centre of common plane} & \frac{3}{8}r & \frac{1}{4}kr \end{array}

[Maximum number: 6]

Figure 5

A thin smooth hollow spherical shell has centre O and radius r. Part of the shell is removed to form a bowl with a plane circular rim. The bowl is fixed with the circular rim uppermost and horizontal. The point A is the lowest point of the bowl, as shown in Figure 5.

The point B is on the rim of the bowl, with OB at an angle $\theta$ to the upward vertical, where $\tan \theta=\frac{12}{5}$
A small ball is placed in the bowl at A. The ball is projected from A with horizontal speed u and moves in the vertical plane AOB. The ball stays in contact with the bowl until it reaches B.

At the instant when the ball reaches B, the speed of the ball is v.
By modelling the ball as a particle and ignoring air resistance,

The point C is such that BC is a diameter of the rim of the bowl.

Given that $u^{2}=4 g r$

use the model to show that, after leaving the inner surface of the bowl at B, the ball falls back into the bowl before reaching C.

BC=2r\sin\theta=\frac{24r}{13}

v^2=u^2-\frac{36gr}{13}=\frac{16gr}{13}

t=\frac{2v\sin\theta}{g}=\frac{24}{13}\sqrt{\frac{r}{g}}

x=v\cos\theta\, t

x=\frac{1920r}{2197}

A1*

$\frac{1920r}{2197}<\frac{24r}{13}=BC$ , so the ball falls back into the bowl before reaching C.

[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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Covered syllabus points

Question 1

Question 2

Question 3

Question (a)

Question (b)

Question (c)