Mechanics — Variant 1

A particle moves in a straight line starting from rest. The displacement $s$ m of the particle from a fixed point $O$ on the line at time $t$ s is given by $s = t^{\frac{5}{2}} - \frac{15}{4}t^{\frac{3}{2}} + 6$. Find the value of $s$ when the particle is again at rest.

A car of mass $1200$ kg is travelling along a straight horizontal road. The power of the car's engine is constant and is equal to $16$ kW. There is a constant resistance to motion of magnitude $500$ N.

Find the acceleration of the car at an instant when its speed is $20$ $\text{m s}^{-1}$.

Assuming that the power and the resistance forces remain unchanged, find the steady speed at which the car can travel. The car comes to the bottom of a straight hill of length $316$ m, inclined at an angle to the horizontal of $\sin^{-1}(\frac{1}{60})$. The power remains constant at $16$ kW, but the magnitude of the resistance force is no longer constant and changes such that the work done against the resistance force in ascending the hill is $128400$ J. The time taken to ascend the hill is $15$ s.

Given that the car is travelling at a speed of $20$ $\text{m s}^{-1}$ at the bottom of the hill, find its speed at the top of the hill.