Practice Paper 1 (Edexcel IAL Maths)

For forces given in vector form, the resultant of the forces is the vector sum of the forces. So for two forces $F_{1}$ and $F_{2}$, for example, the resultant $R$ is given by the vector sum $R=F_{1}+F_{2}$.

A particle is acted upon by two forces, $F_{1}$ and $F_{2}$, given in vector form as

$F_{1}=(14i+3j)N\mathrm{and}F_{2}=(-12i+7j)N$

The resultant of $F_{1}$ and $F_{2}$ is $R$.

Find $R$ and calculate its magnitude.

By drawing a sketch and using trigonometry, show that the angle between $R$ and the vector $i$ is $78.7^{\circ}$ to 3 significant figures.

Two marbles $M$ and $N$ have masses 3 kg and $m$ kg respectively. The marbles are rolled towards each other in a straight line on a smooth horizontal table and they collide directly bringing $M$ to rest. Immediately before the collision, the speeds of $M$ and $N$ are $3ms^{-1}$ and $5ms^{-1}$ respectively. Immediately after the collision, $N$ moves with speed $4ms^{-1}$. The marbles are modelled as particles.

Find the value of $m$.

(i) Find the magnitude of the impulse exerted on $M$ by $N$. Include the units.

(ii) State clearly the direction of this impulse.

Two particles $A$ and $B$ have masses of 5 kg and 9 kg respectively. The particles are connected by a light inextensible string that passes over a smooth light fixed pulley as shown in the diagram below:

The particles are released from rest with the string taut.

Calculate the acceleration of the particles and the tension in the string as $B$ descends.

Two particles A and B, of masses 2.7 kg and 2.2 kg respectively, are connected by means of a light inextensible string. Particle A is held motionless on a rough fixed plane inclined at 25° to the horizontal. The string passes over a smooth light pulley fixed at the top of the plane so that B is hanging vertically downwards as shown in the diagram below:

The string between A and the pulley lies along a line of greatest slope of the plane, and B hangs freely from the pulley. The coefficient of friction between particle A and the plane is μ.

The system is released from rest with the string taut. Given that particle B descends 1.82 m in the first 3 seconds after it is released, find the value of μ.

A particle moves from rest to velocity $7.75ms^{-1}$ in $3.2$ seconds. Find the displacement of the particle.

A train leaves station $O$ from rest with constant acceleration $a = ( 0.3 i + 0.7 j ) m s^{- 2}$. 80 seconds later it passes through (but does not stop at) station $A$ at which point its acceleration changes to $a = ( 0.5 i + 0.3 j ) m s^{- 2}$. 180 seconds later the train passes through station $B$.

Find the displacement of the train from station $O$ when it passes through station $A$.

Find the velocity of the train as it passes through station $A$.

Find the displacement of the train between stations $A$ and $B$.

A particle moves along a horizontal path, its motion, in terms of displacement from point $0$, is shown for the first 20 seconds on the displacement-time graph below. Displacement is measured in metres.

(i)

Write down the distance travelled by the particle in the first 10 seconds.

(ii)

Work out the particle’s average speed in the first 20 seconds.

Convert the following into S.I. units:

(i) $2.7\mathrm{km}$ per hour

(ii) $460\mathrm{km}h^{-2}$

(iii) $3.2\times10^{5}\mathrm{kg} \mathrm{cm}^{-2}$

The velocity of a train is given by $v = ( 55 i - 48 j ) \mathrm{m} \mathrm{s}^{- 1}$. The speed of the train can be calculated by finding the magnitude of the train’s velocity.

(i) Using $s = \left| v \right| = \sqrt{x^{2} + y^{2}}$, find the speed of the train.

(ii) Using $θ=\mathrm{tan}^{-1}(\frac{y}{x})$ , find the angle the direction of motion of the train makes with the unit vector i.

A mathematics teacher has built a decoration for her classroom to celebrate ‘Pi Day’ on March 14^th. The decoration is in the shape of the letter $π$, made of two identical poles and a non-uniform wooden beam $AB$ of mass 1.4 kg and length 4 m. She attaches two identical baskets to the beam, one at point $A$ hanging by a light inextensible string and the other fixed to the beam at point $B$. The two poles are vertical and are placed 2 m apart, and the beam rests horizontally on top of them as shown in the diagram below.

When the beam is placed onto the poles with both baskets empty, and such that the midpoint of the beam is halfway between the two poles, then the beam is at the point of tilting about the pole closest to $A$. With the beam in the same position a mass of 3.2 kg is placed into the basket at $B$, and the beam is then at the point of tilting about the pole closest to $B$.

Use the above information to find the distance of the centre of mass of the beam from $B$, as well as the empty mass of each basket. In your solution you may treat the basket at $B$, along with anything it contains, as being a point mass located at $B$.

Newton’s First Law of Motion states that an object at rest will remain at rest, and an object moving with constant velocity will continue to move with constant velocity, unless an unbalanced force acts on the object.

An avocado with a mass of 170 g is lying motionless on top of a table. State, with a reason, what the total unbalanced force acting on the avocado is.

Lucy and Fran are driving monster trucks along the same straight line in opposite directions towards each other. The combined mass of Lucy and her truck is 5400 kg and the combined mass of Fran and her truck is 4700 kg. They collide directly. Immediately before the collision, Lucy’s truck is moving with speed $4ms^{-1}$ and Fran’s truck is moving with speed $5ms^{-1}$. Immediately after the collision, Lucy and Fran’s monster trucks are moving at the same speed.

(i) Given that Lucy and Fran’s monster trucks are moving in the same direction after the collision, find the speed at which they both travel.

(ii) State which person’s direction of motion has been reversed.

(iii) Calculate the change in momentum of Fran’s monster truck due to the collision.

Given instead that Lucy and Fran’s monster trucks bounce off each other and move in opposite directions with equal speeds after the collision, calculate the speed at which they both travel.