Mechanics — Variant 2

A lorry of mass $16000$ kg is travelling along a straight horizontal road. The engine of the lorry is working at constant power. The work done by the driving force in $10$ s is $750000$ J.

Find the power of the lorry's engine.

There is a constant resistance force acting on the lorry of magnitude $2400$ N. Find the acceleration of the lorry at an instant when its speed is $25$ $\text{m s}^{-1}$.

A particle $P$ of mass $0.4$ kg is on a rough horizontal floor. The coefficient of friction between $P$ and the floor is $\mu$. A force of magnitude $3$ N is applied to $P$ upwards at an angle $\alpha$ above the horizontal, where $\tan \alpha = \frac{3}{4}$. The particle is initially at rest and accelerates at $2$ $\text{m s}^{-2}$.

Find the time it takes for $P$ to travel a distance of $1.44$ m from its starting point.

Find $\mu$.

A cyclist travels along a straight road with constant acceleration. He passes through points $A$, $B$ and $C$. The cyclist takes $2$ seconds to travel along each of the sections $AB$ and $BC$ and passes through $B$ with speed $4.5$ $\text{m s}^{-1}$. The distance $AB$ is $\frac{4}{5}$ of the distance $BC$.

Find the acceleration of the cyclist.

Find $AC$.

On a straight horizontal test track, driverless vehicles (with no passengers) are being tested. A car of mass $1600$ kg is towing a trailer of mass $700$ kg along the track. The brakes are applied, resulting in a deceleration of $12$ $\text{m s}^{-2}$. The braking force acts on the car only. In addition to the braking force there are constant resistance forces of $600$ N on the car and of $200$ N on the trailer.

Find the magnitude of the force in the tow-bar.

Find the braking force.

At the instant when the brakes are applied, the car has speed $22$ $\text{m s}^{-1}$. At this instant the car is $17.5$ m away from a stationary van, which is directly in front of the car. Show that the car hits the van at a speed of $8$ $\text{m s}^{-1}$.

After the collision, the van starts to move with speed $5$ $\text{m s}^{-1}$ and the car and trailer continue moving in the same direction with speed $2$ $\text{m s}^{-1}$. Find the mass of the van.