Mechanics — Variant 1

A car starts from rest and moves in a straight line with constant acceleration for a distance of $200$ m, reaching a speed of $25$ $\text{m s}^{-1}$. The car then travels at this speed for $400$ m, before decelerating uniformly to rest over a period of $5$ s.

Find the time for which the car is accelerating.

Sketch the velocity–time graph for the motion of the car, showing the key points.

Find the average speed of the car during its motion.

Two particles $P$ and $Q$, of masses $0.5$ kg and $0.3$ kg respectively, are connected by a light inextensible string. The string is taut and $P$ is vertically above $Q$. A force of magnitude $10$ N is applied to $P$ vertically upwards. Find the acceleration of the particles and the tension in the string connecting them.

A crate of mass $300$ kg is at rest on rough horizontal ground. The coefficient of friction between the crate and the ground is $0.5$. A force of magnitude $X$ N, acting at an angle $\alpha$ above the horizontal, is applied to the crate, where $\sin \alpha = 0.28$. Find the greatest value of $X$ for which the crate remains at rest.

Two racing cars $A$ and $B$ are at rest alongside each other at a point $O$ on a straight horizontal test track. The mass of $A$ is $1200$ kg. The engine of $A$ produces a constant driving force of $4500$ N. When $A$ arrives at a point $P$ its speed is $25$ $\text{m s}^{-1}$. The distance $OP$ is $d$ m. The work done against the resistance force experienced by $A$ between $O$ and $P$ is $75 000$ J.

Show that $d = 100$. Car $B$ starts off at the same instant as car $A$. The two cars arrive at $P$ simultaneously and with the same speed. The engine of $B$ produces a driving force of $3200$ N and the car experiences a constant resistance to motion of $1200$ N.

Find the mass of $B$.

Find the steady speed which $B$ can maintain when its engine is working at the same rate as it is at $P$.

A particle starts from a point $O$ and moves in a straight line. The velocity $v$ $\text{m s}^{-1}$ of the particle at time $t$ s after leaving $O$ is given by $v = k(3t^{2} - 2t^{3})$, where $k$ is a constant.

Verify that the particle returns to $O$ when $t = 2$.

It is given that the acceleration of the particle is $-13.5$ $\text{m s}^{-2}$ for the positive value of $t$ at which $v = 0$. Find $k$ and hence find the total distance travelled in the first two seconds of motion.

Two particles $A$ and $B$, of masses $0.4$ kg and $0.2$ kg respectively, are moving down the same line of greatest slope of a smooth plane. The plane is inclined at $30^\circ$ to the horizontal, and $A$ is higher up the plane than $B$. When the particles collide, the speeds of $A$ and $B$ are $3$ $\text{m s}^{-1}$ and $2$ $\text{m s}^{-1}$ respectively. In the collision between the particles, the speed of $A$ is reduced to $2.5$ $\text{m s}^{-1}$.

Find the speed of $B$ immediately after the collision. After the collision, when $B$ has moved $1.6$ m down the plane from the point of collision, it hits a barrier and returns back up the same line of greatest slope. $B$ hits the barrier $0.4$ s after the collision, and when it hits the barrier, its speed is reduced by $90\%$. The two particles collide again $0.44$ s after their previous collision, and they then coalesce on impact.

Show that the speed of $B$ immediately after it hits the barrier is $0.5$ $\text{m s}^{-1}$. Hence find the speed of the combined particle immediately after the second collision between $A$ and $B$.