A Level Mechanics Practice Questions

The diagram shows the $(t,\ x)$ graph for a particle which moves along a straight line. Its first straight section runs from $(0,\ -4)$ to the corner at $(2,\ 2)$, and A is the point where the graph first crosses the t-axis. Find the coordinates of the point A.

The diagram shows the $(t,\ x)$ graph for a particle which moves along a straight line, where x metres is the displacement of the particle from a fixed origin. Write down the final displacement of the particle, in metres.

The $(t,\ x)$ graph for a particle moving along a straight line starts at $x=-4$ m when $t=0$ and finishes at $x=-2$ m when $t=7$ s, where x metres is the displacement from a fixed origin. Find the average velocity of the particle, in ms$^{-1}$.

A particle is projected from the point O with speed 8 ms$^{-1}$ at an angle $\theta$ above the horizontal, and it passes through the point $A(8;\ -1.81)$, distances in metres, with O as origin. Taking $g=9.81$ ms$^{-2}$, there are two possible values of $\theta$. Find the smaller one, in degrees, correct to 1 decimal place.

A particle is projected from O with speed 8 ms$^{-1}$ at an angle $\theta$ above the horizontal and passes through the point $A(8;\ -1.81)$, distances in metres. The two possible values of $\theta$ are $45^{\circ}$ and $32.2^{\circ}$. Which value gives the minimum time taken to reach A, and why?

A particle A of mass $6m$ kg rests on a rough plane inclined at $30^{\circ}$ to the horizontal, with coefficient of friction 0.25. A light inextensible string runs from A over a smooth pulley at the top of the plane to a particle B of mass $2m$ kg hanging freely. The system is released from rest. Does A slide down the plane?

A light inextensible string carrying tension T passes over a smooth pulley P fixed at the top of a plane inclined at $30^{\circ}$ to the horizontal. One part of the string runs down the slope to a particle on the plane, and the other hangs vertically. What is the magnitude of the force exerted by the string on P?

A man pulls a crate of weight 20 N along a rough horizontal floor using a string inclined at 60° to the horizontal. The crate is in limiting equilibrium when he pulls with a force of magnitude 4 N. Find the exact normal reaction between the crate and the floor.

Continuing the crate scenario (weight 20 N, pulling force 4 N at 60° to the horizontal, normal reaction $R=20-2\sqrt3$ N, limiting equilibrium), find the exact coefficient of friction between the crate and the floor.

A car travels at 40 ms$^{-1}$ and decelerates uniformly to 25 ms$^{-1}$ over 30 seconds, then continues at the constant speed of 25 ms$^{-1}$ for a further distance of 500 m. Find the total distance travelled by the car.

For the same journey (uniform deceleration from 40 to 25 ms$^{-1}$ over the first 30 s, then constant 25 ms$^{-1}$ for the next 500 m), which statement correctly describes the shape of the displacement-time $(t,x)$ graph?

A toy car P (mass 4 kg) rests on a rough horizontal plane, with coefficient of friction $\frac12$ between P and the plane, and is connected by a light inextensible string over a smooth pulley X to toy car Q (mass 8 kg) held on a smooth plane inclined at 30° to the horizontal. When Q is released, find the acceleration of the system and the tension in the string, in terms of $g$.

A ring of mass 0.35 kg is threaded on a horizontal wire and moves at a uniform speed under a horizontal force of 2.4 N applied parallel to the wire. Taking $g=10\text{ ms}^{-2}$, what is the coefficient of friction between the ring and the wire?

A block of mass 3.5 kg is released from rest at point A on a plane inclined at angle $\alpha$ to the horizontal, where $\tan\alpha = \dfrac{3}{4}$. It slides down to point B at the base of the plane. The coefficient of friction between the block and the plane is $\dfrac{1}{4}$. Take $g = 9.81$ ms$^{-2}$.

Four coplanar forces act at the point O, as shown in the diagram. A force of 2N acts along the positive y-axis. A force of 3N acts at 30 degrees to the left of the positive y-axis. A force of 5N acts at 30 degrees to the right of the positive y-axis. A fourth force of 2N acts at right angles to the 5N force, below the positive x-axis. The forces $\mathbf{i}$ and $\mathbf{j}$ are of 1N magnitude in the directions of the x and y axes respectively.

A stone is projected from a point A on level ground with a speed of 20 ms$^{-1}$ at an angle of 30° to the horizontal, towards a pole standing 30 m away. Find the time, in seconds, at which the stone is level with the pole. Give your answer correct to 3 significant figures.

A particle is projected at an angle $\theta$ above the horizontal where $\theta=\sin^{-1}\left(\dfrac{\sqrt{2}}{2}\right)$. Find $\theta$ in degrees.

Two forces have magnitudes $P$ and $Q$ and the acute angle between them is $\theta$, so their resultant has magnitude $R$ where $R^{2}=P^{2}+Q^{2}+2PQ\cos\theta$. Two forces of magnitudes 5 N and 8 N act at an angle of $60^{\circ}$ to each other. Find the magnitude of their resultant, in newtons, correct to 3 significant figures.

A car passes a fixed point A at 10 ms$^{-1}$ and holds that velocity for $t_{1}$ seconds. It then accelerates uniformly over the next $t_{2}$ seconds until it reaches 15 ms$^{-1}$.

Two concurrent coplanar forces of magnitude 4 N and 9 N act on a particle. The angle between the two forces is 40 degrees. Find the magnitude of their resultant.

A particle of mass 8 kg rests on a rough plane inclined at 30$^{\circ}$ to the horizontal and is on the point of slipping. Take $g = 9.81$ ms$^{-2}$.

Particles A and B of masses 5 kg and 8 kg respectively are connected by a light inextensible string passing over a smooth pulley. Particle A lies on a smooth plane inclined at an angle $\tan^{-1}\left(\dfrac{4}{3}\right)$ to the horizontal and particle B hangs freely. The system is released from rest. Take $g = 9.81$ ms$^{-2}$.

The trajectory of a projectile is described by $y = 2x - 0.01x^{2}$, where x is the horizontal displacement and y is the vertical displacement from the point of projection.

A particle is projected horizontally from a point O at a height of 45 m vertically above a point P on level ground. It hits the ground at a point Q with PQ = 15 m. Take $g = 9.81$ ms$^{-2}$.