Mechanics Paper 4 June 2012

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$^{-1}$.

Write down an expression for the magnitude of the car's acceleration during that stage, in terms of $t_{2}$.

A car passes a fixed point A at 10 ms$^{-1}$, holds that velocity for a while, then accelerates uniformly over $t_{2}$ seconds until it reaches 15 ms$^{-1}$, and finally decelerates uniformly to rest in a further $t_{3}$ seconds. The magnitudes of the acceleration and the deceleration are equal.

Given that $t_{2} = 4$ seconds, find $t_{3}$.

A ball is projected from the top of a building 40 m high with an initial velocity of 20 ms$^{-1}$ at an angle of 30$^{\circ}$ to the horizontal, and hits the ground at a point P. Take $g = 9.81$ ms$^{-2}$.

Find the time of flight of the ball.

A ball is projected from the top of a building 40 m high with an initial velocity of 20 ms$^{-1}$ at an angle of 30$^{\circ}$ to the horizontal, and hits the ground at a point P after a flight of 4.052 seconds. Take $g = 9.81$ ms$^{-2}$.

Find the angle, to the nearest degree, that the ball's direction of motion at P makes with the horizontal.

Two particles of mass $m_{1}$ kg and $m_{2}$ kg, where $m_{1} > m_{2}$, are connected by a light inelastic string passing over a smooth fixed pulley, and are released from rest.

Which expression gives the acceleration of the system?

Two particles of mass $m_{1}$ kg and $m_{2}$ kg, where $m_{1} > m_{2}$, are connected by a light inelastic string passing over a smooth fixed pulley. The tension in the string is $\dfrac{2m_{1}m_{2}g}{m_{1}+m_{2}}$.

Calculate the tension, in newtons, when $m_{1} = 5$ kg and $m_{2} = 3$ kg, taking $g = 9.81$ ms$^{-2}$.

Two particles hang from the ends of a light inelastic string that passes over a smooth fixed pulley.

State what is true of the magnitude of the tension at the two ends of the string.

A particle of weight 20 N rests on a rough horizontal surface. A force P N inclined at an angle $\theta$ to the horizontal is applied to the particle until it is on the point of moving. The normal force on the particle is $R = 16j$ and the frictional force is $F = -9i$, where $i$ and $j$ are horizontal and vertical unit vectors.

Calculate the coefficient of friction.

A particle of weight 20 N rests on a rough horizontal surface. A force P N inclined at an angle $\theta$ to the horizontal is applied to the particle until it is on the point of moving. The normal force on the particle is $R = 16j$ and the frictional force is $F = -9i$, where $i$ and $j$ are horizontal and vertical unit vectors.

Calculate the magnitude of the total contact force between the particle and the surface.

A particle of weight 20 N rests on a rough horizontal surface. A force P N inclined at an angle $\theta$ to the horizontal is applied to the particle until it is on the point of moving. The normal force on the particle is $R = 16j$ and the frictional force is $F = -9i$, where $i$ and $j$ are horizontal and vertical unit vectors.

Calculate $\theta$, giving your answer to the nearest degree.

A particle of weight 20 N rests on a rough horizontal surface. A force P N inclined at an angle $\theta$ to the horizontal is applied to the particle until it is on the point of moving. The normal force on the particle is $R = 16j$ and the frictional force is $F = -9i$, where $i$ and $j$ are horizontal and vertical unit vectors.

Calculate P.