Questions - Edexcel - A-Level Maths

Show that the time of flight from $O$ to $T$ is $\sqrt{\left(\frac{78a}{g}\right)}$. [3]
Find the speed of the particle immediately before it hits $T$. [3]

Edexcel M2 2013 June Q1

8 marks Moderate -0.3

Three particles of masses 2 kg, 3 kg and $m$ kg are positioned at the points with coordinates $(a, 3)$, $(3, -1)$ and $(-2, 4)$ respectively. Given that the centre of mass of the particles is at the point with coordinates $(0, 2)$, find

the value of $m$, [4]
the value of $a$. [4]

Edexcel M2 2013 June Q2

7 marks Moderate -0.3

A car has mass 1200 kg. The maximum power of the car's engine is 32 kW. The resistance to motion due to non-gravitational forces is modelled as a force of constant magnitude 800 N. When the car is travelling on a horizontal road at constant speed $V$ m s$^{-1}$, the engine of the car is working at maximum power.

Find the value of $V$. [3]

The car now travels downhill on a straight road inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac{1}{40}$. The resistance to motion due to non-gravitational forces is still modelled as a force of constant magnitude 800 N. Given that the engine of the car is again working at maximum power,

find the acceleration of the car when its speed is 20 m s$^{-1}$. [4]

Edexcel M2 2013 June Q3

13 marks Moderate -0.3

A particle $P$ of mass 0.25 kg moves under the action of a single force $\mathbf{F}$ newtons. At time $t$ seconds, the velocity of $P$ is $\mathbf{v}$ m s$^{-1}$, where $$\mathbf{v} = (2 - 4t)\mathbf{i} + (t^2 + 2t)\mathbf{j}$$ When $t = 0$, $P$ is at the point with position vector $(2\mathbf{i} - 4\mathbf{j})$ m with respect to a fixed origin $O$. When $t = 3$, $P$ is at the point $A$. Find

the momentum of $P$ when $t = 3$, [2]
the magnitude of $\mathbf{F}$ when $t = 3$, [6]
the position vector of $A$. [5]

Edexcel M2 2013 June Q4

10 marks Standard +0.3

\includegraphics{figure_1} The points $O$ and $B$ are on horizontal ground. The point $A$ is $h$ metres vertically above $O$. A particle $P$ is projected from $A$ with speed 12 m s$^{-1}$ at an angle $\alpha°$ to the horizontal. The particle moves freely under gravity and hits the ground at $B$, as shown in Figure 1. The speed of $P$ immediately before it hits the ground is 15 m s$^{-1}$.

By considering energy, find the value of $h$. [4]

Given that 1.5 s after it is projected from $A$, $P$ is at a point 4 m above the level of $A$, find

the value of $\alpha$, [3]
the direction of motion of $P$ immediately before it reaches $B$. [3]

Edexcel M2 2013 June Q5

12 marks Standard +0.3

\includegraphics{figure_2} The uniform L-shaped lamina $OABCDE$, shown in Figure 2, is made from two identical rectangles. Each rectangle is 4 metres long and $a$ metres wide. Giving each answer in terms of $a$, find the distance of the centre of mass of the lamina from

$OE$, [4]
$OA$. [4]

The lamina is freely suspended from $O$ and hangs in equilibrium with $OE$ at an angle $\theta$ to the downward vertical through $O$, where $\tan \theta = \frac{4}{3}$.

Find the value of $a$. [4]

Edexcel M2 2013 June Q6

12 marks Standard +0.3

\includegraphics{figure_3} A uniform rod $AB$ has weight 30 N and length 3 m. The rod rests in equilibrium on a rough horizontal peg $P$ with its end $A$ on smooth horizontal ground. The rod is in a vertical plane perpendicular to the peg. The rod is inclined at 15° to the ground and the point of contact between the peg and the rod is 45 cm above the ground, as shown in Figure 3.

Show that the normal reaction at $P$ has magnitude 25 N. [4]
Find the magnitude of the force on the rod at $A$. [4]

The coefficient of friction between the rod and the peg is $\mu$.

Find the range of possible values of $\mu$. [4]

Edexcel M2 2013 June Q7

13 marks Standard +0.3

\includegraphics{figure_4} Two smooth particles $P$ and $Q$ have masses $m$ and $2m$ respectively. The particles are moving in the same direction in the same straight line, on a smooth horizontal plane, with $Q$ in front of $P$. The particles are moving towards a fixed smooth vertical wall which is perpendicular to the direction of motion of the particles, as shown in Figure 4. The speed of $P$ is $2u$ and the speed of $Q$ is $3u$. The coefficient of restitution between $Q$ and the wall is $\frac{1}{3}$. Particle $Q$ strikes the wall, rebounds and then collides directly with $P$. The direction of motion of each particle is reversed by this collision. Immediately after this collision the speed of $P$ is $v$ and the speed of $Q$ is $w$.

Show that $v = 2w$. [5]

The total kinetic energy of $P$ and $Q$ immediately after they collide is half the total kinetic energy of $P$ and $Q$ immediately before they collide.

Find the coefficient of restitution between $P$ and $Q$. [8]

Edexcel M3 2016 June Q1

8 marks Standard +0.3

A particle is attached to one end of a light inextensible string of length $l$. The other end of the string is attached to a fixed point $A$. The particle moves with constant angular speed $\omega$ in a horizontal circle. The centre of the circle is vertically below $A$ and the radius of the circle is $r$. Show that $\omega^2 = \frac{g}{\sqrt{l^2 - r^2}}$ [8]

Edexcel M3 2016 June Q2

9 marks Standard +0.3

A light elastic spring, of natural length $5a$ and modulus of elasticity $10mg$, has one end attached to a fixed point $A$ on a ceiling. A particle $P$ of mass $2m$ is attached to the other end of the spring and $P$ hangs freely in equilibrium at the point $O$.

Find the distance $AO$. [3]

The particle is now pulled vertically downwards a distance $\frac{1}{2}a$ from $O$ and released from rest.

Show that $P$ moves with simple harmonic motion. [4]
Find the period of the motion. [2]

Edexcel M3 2016 June Q3

7 marks Standard +0.8

A particle $P$ of mass $m$ is attached to one end of a light elastic string, of natural length $l$ and modulus of elasticity $4mg$. The other end of the string is attached to a fixed point $O$ on a rough horizontal plane. The coefficient of friction between $P$ and the plane is $\frac{2}{5}$. The particle is held at a point $A$ on the plane, where $OA = \frac{5}{4}l$, and released from rest. The particle comes to rest at the point $B$.

Show that $OB < l$ [4]
Find the distance $OB$. [3]

Edexcel M3 2016 June Q4

9 marks Standard +0.8

A particle $P$ of mass $m$ is fired vertically upwards from a point on the surface of the Earth and initially moves in a straight line directly away from the centre of the Earth. When $P$ is at a distance $x$ from the centre of the Earth, the gravitational force exerted by the Earth on $P$ is directed towards the centre of the Earth and has a magnitude which is inversely proportional to $x^2$. At the surface of the Earth the acceleration due to gravity is $g$. The Earth is modelled as a fixed sphere of radius $R$.

Show that the magnitude of the gravitational force acting on $P$ is $\frac{mgR^2}{x^2}$ [2]

The particle was fired with initial speed $U$ and the greatest height above the surface of the Earth reached by $P$ is $\frac{R}{20}$. Given that air resistance can be ignored,

find $U$ in terms of $g$ and $R$. [7]

Edexcel M3 2016 June Q5

11 marks Standard +0.8

A vertical ladder is fixed to a wall in a harbour. On a particular day the minimum depth of water in the harbour occurs at 0900 hours. The next time the water is at its minimum depth is 2115 hours on the same day. The bottom step of the ladder is 1 m above the lowest level of the water and 9 m below the highest level of the water. The rise and fall of the water level can be modelled as simple harmonic motion and the thickness of the step can be assumed to be negligible. Find

the speed, in metres per hour, at which the water level is moving when it reaches the bottom step of the ladder, [7]
the length of time, on this day, between the water reaching the bottom step of the ladder and the ladder being totally out of the water once more. [4]

Edexcel M3 2016 June Q6

14 marks Standard +0.8

\includegraphics{figure_1} A smooth solid hemisphere of radius 0.5 m is fixed with its plane face on a horizontal floor. The plane face has centre $O$ and the highest point of the surface of the hemisphere is $A$. A particle $P$ has mass 0.2 kg. The particle is projected horizontally with speed $u$ m s$^{-1}$ from $A$ and leaves the hemisphere at the point $B$, where $OB$ makes an angle $\theta$ with $OA$, as shown in Figure 1. The point $B$ is at a vertical distance of 0.1 m below the level of $A$. The speed of $P$ at $B$ is $v$ m s$^{-1}$

Show that $v^2 = u^2 + 1.96$ [3]
Find the value of $u$. [4]

The particle first strikes the floor at the point $C$.

Find the length of $OC$. [7]

Edexcel M3 2016 June Q7

17 marks Challenging +1.2

Use algebraic integration to show that the centre of mass of a uniform solid right circular cone of height $h$ is at a distance $\frac{3}{4}h$ from the vertex of the cone. [You may assume that the volume of a cone of height $h$ and base radius $r$ is $\frac{1}{3}\pi r^2 h$] [5]

\includegraphics{figure_2} A uniform solid $S$ consists of a right circular cone, of radius $r$ and height $5r$, fixed to a hemisphere of radius $r$. The centre of the plane face of the hemisphere is $O$ and this plane face coincides with the base of the cone, as shown in Figure 2.