Questions - Edexcel - A-Level Maths

the magnitude of the impulse given to the ball,
the size of the angle that the direction of this impulse makes with the original direction of motion $AB$.

[8]

Edexcel M2 2010 January Q5

11 marks Standard +0.3

A cyclist and her bicycle have a total mass of $70$ kg. She cycles along a straight horizontal road with constant speed $3.5 \text{ ms}^{-1}$. She is working at a constant rate of $490$ W.

Find the magnitude of the resistance to motion. [4]

The cyclist now cycles down a straight road which is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac{1}{14}$, at a constant speed $U \text{ ms}^{-1}$. The magnitude of the non-gravitational resistance to motion is modelled as $40U$ newtons. She is now working at a constant rate of $24$ W.

Find the value of $U$. [7]

Edexcel M2 2010 January Q6

7 marks Standard +0.3

\includegraphics{figure_2} A uniform rod $AB$, of mass $20$ kg and length $4$ m, rests with one end $A$ on rough horizontal ground. The rod is held in limiting equilibrium at an angle $\alpha$ to the horizontal, where $\tan \alpha = \frac{3}{4}$, by a force acting at $B$, as shown in Figure 2. The line of action of this force lies in the vertical plane which contains the rod. The coefficient of friction between the ground and the rod is $0.5$. Find the magnitude of the normal reaction of the ground on the rod at $A$. [7]

Edexcel M2 2010 January Q7

11 marks Standard +0.8

[The centre of mass of a semi-circular lamina of radius $r$ is $\frac{4r}{3\pi}$ from the centre] \includegraphics{figure_3} A template $T$ consists of a uniform plane lamina $PQRQS$, as shown in Figure 3. The lamina is bounded by two semicircles, with diameters $SQ$ and $QR$, and by the sides $SP$, $PQ$ and $QR$ of the rectangle $PQRS$. The point $O$ is the mid-point of $SR$, $PQ = 12$ cm and $QR = 2$ cm.

Show that the centre of mass of $T$ is a distance $\frac{4|2x^2 - 3|}{8x + 3\pi}$ cm from $SR$. [7]

The template $T$ is freely suspended from the point $P$ and hangs in equilibrium. Given that $x = 2$ and that $\theta$ is the angle that $PQ$ makes with the horizontal,

show that $\tan \theta = \frac{48 + 9\pi}{22 + 6\pi}$. [4]

Edexcel M2 2010 January Q8

17 marks Standard +0.3

[In this question $\mathbf{i}$ and $\mathbf{j}$ are unit vectors in a horizontal and upward vertical direction respectively] A particle $P$ is projected from a fixed point $O$ on horizontal ground with velocity $u(\mathbf{i} + c\mathbf{j}) \text{ ms}^{-1}$, where $c$ and $u$ are positive constants. The particle moves freely under gravity until it strikes the ground at $A$, where it immediately comes to rest. Relative to $O$, the position vector of a point on the path of $P$ is $(x\mathbf{i} + y\mathbf{j})$ m.

Show that $$y = cx - \frac{4.9x^2}{u^2}.$$ [5]

Given that $u = 7$, $OA = R$ m and the maximum vertical height of $P$ above the ground is $H$ m,

using the result in part (a), or otherwise, find, in terms of $c$,
1. $R$
2. $H$.
[6]

Given also that when $P$ is at the point $Q$, the velocity of $P$ is at right angles to its initial velocity,

find, in terms of $c$, the value of $x$ at $Q$. [6]

Edexcel M2 2012 January Q1

4 marks Easy -1.2

A tennis ball of mass $0.1$ kg is hit by a racquet. Immediately before being hit, the ball has velocity $30\mathbf{i}$ m s$^{-1}$. The racquet exerts an impulse of $(-2\mathbf{i} - 4\mathbf{j})$ N s on the ball. By modelling the ball as a particle, find the velocity of the ball immediately after being hit. [4]

Edexcel M2 2012 January Q2

10 marks Moderate -0.3

A particle $P$ is moving in a plane. At time $t$ seconds, $P$ is moving with velocity $\mathbf{v}$ m s$^{-1}$, where $\mathbf{v} = 2t\mathbf{i} - 3t^2\mathbf{j}$. Find

the speed of $P$ when $t = 4$ [2]
the acceleration of $P$ when $t = 4$ [3]

Given that $P$ is at the point with position vector $(-4\mathbf{i} + \mathbf{j})$ m when $t = 1$,

find the position vector of $P$ when $t = 4$ [5]

Edexcel M2 2012 January Q3

10 marks Standard +0.3

A cyclist and her cycle have a combined mass of $75$ kg. The cyclist is cycling up a straight road inclined at $5°$ to the horizontal. The resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude $20$ N. At the instant when the cyclist has a speed of $12$ m s$^{-1}$, she is decelerating at $0.2$ m s$^{-2}$.

Find the rate at which the cyclist is working at this instant. [5]

When the cyclist passes the point $A$ her speed is $8$ m s$^{-1}$. At $A$ she stops working but does not apply the brakes. She comes to rest at the point $B$. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude $20$ N.

Use the work-energy principle to find the distance $AB$. [5]

Edexcel M2 2012 January Q4

10 marks Standard +0.3

\includegraphics{figure_1} The trapezium $ABCD$ is a uniform lamina with $AB = 4$ m and $BC = CD = DA = 2$ m, as shown in Figure 1.

Show that the centre of mass of the lamina is $\frac{4\sqrt{3}}{9}$ m from $AB$. [5]

The lamina is freely suspended from $D$ and hangs in equilibrium.

Find the angle between $DC$ and the vertical through $D$. [5]

Edexcel M2 2012 January Q5

11 marks Standard +0.3

\includegraphics{figure_2} A uniform rod $AB$ has mass $4$ kg and length $1.4$ m. The end $A$ is resting on rough horizontal ground. A light string $BC$ has one end attached to $B$ and the other end attached to a fixed point $C$. The string is perpendicular to the rod and lies in the same vertical plane as the rod. The rod is in equilibrium, inclined at $20°$ to the ground, as shown in Figure 2.

Find the tension in the string. [4]

Given that the rod is about to slip,

find the coefficient of friction between the rod and the ground. [7]

Edexcel M2 2012 January Q6

15 marks Standard +0.3

Three identical particles, $A$, $B$ and $C$, lie at rest in a straight line on a smooth horizontal table with $B$ between $A$ and $C$. The mass of each particle is $m$. Particle $A$ is projected towards $B$ with speed $u$ and collides directly with $B$. The coefficient of restitution between each pair of particles is $\frac{2}{3}$.

Find, in terms of $u$,
1. the speed of $A$ after this collision,
2. the speed of $B$ after this collision.
[7]
Show that the kinetic energy lost in this collision is $\frac{5}{36}mu^2$ [4]

After the collision between $A$ and $B$, particle $B$ collides directly with $C$.