Practice Paper 2 (Edexcel IAL Maths)

The normal reaction is the force that acts in a direction perpendicular to a surface when an object is in contact with the surface. In practical terms, it is what stops things from falling straight through solid objects!

A large maths textbook with a weight of 25 N is sitting at rest on a horizontal table. State, with a reason, the size and direction of the normal reaction force acting on the book.

Amber works at a circus and routinely performs the human cannonball act. The act involves Amber, mass 65 kg, being ejected horizontally out of a specially designed cannon, mass 300 kg, into a net 100 metres away. Both the cannon and Amber start from rest before Amber is shot out at a speed of 108 km h^-1.

(i) Find the speed of the cannon after ejecting Amber.

(ii) Show that the cannon recoils after ejecting Amber.

Amber is travelling with a horizontal speed of 26 m s^-1 when she hits the net. The horizontal resistance force is assumed to be constant.

Find the time it takes Amber to reach the net after being ejected from the canon.

Calculate the magnitude of the resistance force.

A train locomotive of mass 7000 kg and a carriage of mass 2000 kg are at rest on a section of horizontal track. The connection between the locomotive and carriage may be modelled as a light rod parallel to the direction of their motion forward or backward along the track. The resistances to motion of the locomotive and the carriage are modelled as constant forces of 2300 N and 1000 N respectively.

The locomotive begins to accelerate in the backwards direction, with its engine providing a constant driving force of 15 000 N.

Find the magnitude of the acceleration of the locomotive and carriage.

Find the thrust in the connecting rod.

A particle of mass *m *kg hangs in equilibrium, suspended by two light inextensible strings. The strings are inclined at 25° and 70° to the horizontal, as shown in the diagram below:

Given that the tension in the string angled at 70° to the horizontal is 56 N, find the value of m.

A particle passes a fixed point, $0$, with velocity $7.3ms^{-1}$ and then decelerates at a constant $0.32ms^{-2}$. Find the velocity of the particle when its displacement from $0$ is $23m$. Give your answer to three significiant figures.

A particle moves under constant acceleration $[ ( 2 q + 1 ) i + ( q - 1 ) j ] m s^{ - } 2$. In a 6 second time period, the particle has initial velocity $[ 5.2 i + ( 1 - 2 p ) j ] m s^{ - } 1$ and final velocity $[ ( 27 p + 4 ) i + 5.2 j ] m s^{ - } 1$ where $p$ and $q$ are constants.

Show that $27p-12q=7.2\mathrm{and}2p-6q=-10.2$

Find the values of $p$ and $q$.

A particle is set into motion with a constant velocity of $3ms^{-1}$. After 5 seconds the particle stops and remains stationary for 6 seconds. The particle then moves with a constant velocity of $-6ms^{-1}$ until it returns to its initial position.

Sketch a displacement-time graph for the motion of the particle.

Define each of the following and give an example of how they could be used in a mathematical model, include any related assumptions which can be made.

A lamina and a non-uniform rod.

A bead and wire.

Given that find the speed, s equals open vertical bar v close vertical bar, find the speed, s of the following particles.

(i) A particle moving at a velocity of $( - 6 i - j ) \text{cm}$ per minute.

(ii) A particle moving at a velocity of $( 8 i - 12 j ) \mathrm{m} \mathrm{s}^{- 1}$.

(iii) A particle moving at a velocity of $( 40 i + 180 j ) \text{km} \mathrm{h}^{- 1}$.

Using your answers from part (a), or otherwise, calculate the distance each particle will have travelled in 3 minutes. Be careful with units throughout your calculations.

$AB$ is a uniform rod of length 1.7 m and weight 50 N. $AB$ rests horizontally on supports placed at points $C$ and $D$, with $AC=0.4m,CD=1m\mathrm{and}DB=0.3m$, as shown in the diagram below:

Calculate the magnitude of the reaction force at each of the support points.

Given that the particle in each of the following diagrams is in equilibrium, work out the value of $P$:

A particle $P$ has mass 8 kg and a particle $Q$ has mass 5 kg. $P$ is moving towards $Q$ in a straight line on a smooth horizontal surface. The particles collide directly.

Immediately before the collision the speed of $P$ is 3 m s^-1 and $Q$ is at rest.

Immediately after the collision $P$ is travelling in the same direction with speed 1 m s^-1.

Use the principle of conservation of linear momentum to find the speed of $Q$ immediately after the collision.

Show that the magnitude of the impulse exerted by $Q$ on $P$ in the collision is 16 N s.

Use the impulse-momentum principle to show that the impulse exerted by $P$ on $Q$ in the collision has the same magnitude as the impulse exerted by $Q$ on $P$ and that it is acting in the opposite direction.

In the mystical kingdom of Newtonia, a unicorn is using its horn to push a large stone of power across the icy ground towards the spot known as the Point of Destiny. The mass of the stone is 4600 kg, and the ground is perfectly level. As the stone moves across the ground it experiences a constant resistance to movement of 3200 N. The unicorn’s horn, which may be modelled as a light rod, is held horizontal at all times. While the unicorn’s magic would allow it to push with almost any required force, the maximum thrust which the unicorn’s horn may withstand without shattering is 100 kN.

When the stone is exactly 100 metres away from the Point of Destiny and is being pushed at a speed of 1.8 metres per second, an evil wizard appears and begins to cast a spell of doom. The spell will take exactly 3 seconds for the wizard to cast.

Given that the acceleration of the stone must remain constant over the entire 100 metre distance, and that the unicorn’s horn must not be allowed to shatter, determine whether or not the unicorn can get the stone to the Point of Destiny before the wizard completes his spell. Be sure to show complete mathematical workings to support your answer.

For an object on a rough surface, the force of friction will always act to oppose the motion of the object along the surface. The maximum frictional force, $F_{MAX}$, may be found by the formula

$F_{MAX}=\mathrm{μR}$

where µ is a constant called the coefficient of friction, and $R$ is the normal reaction force exerted by the surface on the object.

If the object is at rest and the sum of other forces acting on the object parallel to the surface is less than or equal to $F_{MAX}$, then the object will remain at rest. In that case the actual frictional force will be equal in magnitude and opposite in direction to the sum of the other forces. If the sum of the other forces in the parallel direction is greater than $F_{MAX}$, then the object will begin to accelerate in the direction of those other forces. In that case the force of friction will be equal to $F_{MAX}$, and it will act in the direction opposite to the direction of motion of the object.

For an object in motion, the force of friction will always be equal to $F_{MAX}$, and will act in the direction opposite to the direction of motion of the object.

In each of the situations depicted below, the object is initially at rest. Find the magnitude of the frictional force F that will act upon the object in each case, and determine whether the object will remain at rest or begin to move.

(i)

(ii)

(iii)

(iv)

A particle is projected upwards from ground level with velocity $ums^{-1}.$ 6 seconds later it has a velocity of $1.2ms^{-1}$. Find the value of u.

Find the displacement of the particle, from its initial position, 4 seconds later.