Mechanics — Variant 1

A winch operates by means of a force applied by a rope. The winch is used to pull a load of mass $50$ kg up a line of greatest slope of a plane inclined at $60^\circ$ to the horizontal. The winch pulls the load a distance of $5$ m up the plane at constant speed. There is a constant resistance to motion of $100$ N. Find the work done by the winch.

Two cyclists, Isabella and Maria, are having a race. They both travel along a straight road with constant acceleration, starting from rest at point $A$. Isabella accelerates for $5$ s at a constant rate $a$ $\text{m s}^{-2}$. She then travels at the constant speed she has reached for $10$ s, before decelerating to rest at a constant rate over a period of $5$ s. Maria accelerates at a constant rate, reaching a speed of $5$ $\text{m s}^{-1}$ in a distance of $27.5$ m. She then maintains this speed for a period of $10$ s, before decelerating to rest at a constant rate over a period of $5$ s.

Given that $a = 1.1$, find which cyclist travels further.

Find the value of $a$ for which the two cyclists travel the same distance.

A particle moving in a straight line starts from rest at a point $A$ and comes instantaneously to rest at a point $B$. The acceleration of the particle at time $t$ s after leaving $A$ is $a$ $\text{m s}^{-2}$, where $a = 6t^{\frac{1}{2}} - 2t$.

Find the value of $t$ at point $B$.

Find the distance travelled from $A$ to the point at which the acceleration of the particle is again zero.