Mechanics — Variant 2

A particle is projected vertically upwards from horizontal ground. The speed of the particle $2$ seconds after it is projected is $5 \text{m s}^{-1}$ and it is travelling downwards.

Find the speed of projection of the particle.

Find the distance travelled by the particle between the two times at which its speed is $10 \text{m s}^{-1}$.

A crate of mass $600$ kg is being pulled up a line of greatest slope of a rough plane at a constant speed of $2 \text{m s}^{-1}$ by a rope attached to a winch. The plane is inclined at an angle of $30^\circ$ to the horizontal and the rope is parallel to the plane. The winch is working at a constant rate of $8$ kW. Find the coefficient of friction between the crate and the plane.

A particle moves in a straight line starting from a point $O$. The velocity $v \text{m s}^{-1}$ of the particle $t$ s after leaving $O$ is given by $v = t^{3} - \frac{9}{2}t^{2} + 1$ for $0 \leqslant t \leqslant 4$. You may assume that the velocity of the particle is positive for $t < \frac{1}{2}$, is zero at $t = \frac{1}{2}$ and is negative for $t > \frac{1}{2}$.

Find the distance travelled between $t = 0$ and $t = \frac{1}{2}$.

Find the positive value of $t$ at which the acceleration is zero. Hence find the total distance travelled between $t = 0$ and this instant.

A car of mass $1800$ kg is towing a trailer of mass $300$ kg up a straight road inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = 0.05$. The car and trailer are connected by a tow-bar which is light and rigid and is parallel to the road. There is a resistance force of $800$ N acting on the car and a resistance force of $F$ N acting on the trailer. The driving force of the car's engine is $3000$ N.

It is given that $F = 100$. Find the acceleration of the car and the tension in the tow-bar.

It is given instead that the total work done against $F$ in moving a distance of $50$ m up the road is $6000$ J. The speed of the car at the start of the $50$ m is $20 \text{m s}^{-1}$. Use an energy method to find the speed of the car at the end of the $50$ m.