Mechanics — Variant 1

A car starts from rest and accelerates at $2$ $\text{m s}^{-2}$ for $10$ s. It then travels at a constant speed for $30$ s. The car then uniformly decelerates to rest over a period of $20$ s.

Sketch a velocity-time graph for the motion of the car.

Find the total distance travelled by the car.

A train of mass $180000$ kg ascends a straight hill of length $1.5$ km, inclined at an angle of $1.5^\circ$ to the horizontal. As it ascends the hill, the total work done to overcome the resistance to motion is $12000$ kJ and the speed of the train decreases from $45$ $\text{m s}^{-1}$ to $40$ $\text{m s}^{-1}$. Find the work done by the engine of the train as it ascends the hill, giving your answer in kJ.

A car of mass $1700$ kg is pulling a trailer of mass $300$ kg along a straight horizontal road. The car and trailer are connected by a light inextensible cable which is parallel to the road. There are constant resistances to motion of $400$ N on the car and $150$ N on the trailer. The power of the car's engine is $14000$ W. Find the acceleration of the car and the tension in the cable when the speed is $20$ $\text{m s}^{-1}$.

A straight slope of length $60$ m is inclined at an angle of $12^\circ$ to the horizontal. A bobsled starts at the top of the slope with a speed of $5$ $\text{m s}^{-1}$. The bobsled slides directly down the slope.

It is given that there is no resistance to the bobsled's motion. Find its speed when it reaches the bottom of the slope.

It is given instead that the coefficient of friction between the bobsled and the slope is $0.03$. Find the time that it takes for the bobsled to reach the bottom of the slope.