If a particle speeds up then energy must be given to the particle to increase it's kinetic energy. If it is moving up a slope then some of the energy supplied must go to increase the potential energy of the particle, and if there is any resistive force then some of the supplied energy must be used to overcome this force. In general we can apply one of the equations:

Energy Supplied=Increase in Kinetic Energy +Work Done Against Friction+Increase in Potential Energy

whereis the velocity,is the acceleration,is the resistive force andis the angle if any that the slope makes with the horizontal.

Example: A cyclist travels along a straight road working at a constant rate of 420 W. The total mass of the cyclist and her cycle is 75 kg. Ignoring any resistance to motion, find the acceleration of the cyclist at an instant when she is travelling at

i)given that the road is horizontal,

ii)given that the road is inclined at 1.5◦ to the horizontal and the cyclist is travelling up the slope.

i)

ii)

Example: A car of mass 1200 kg travels along a horizontal straight road. The power of the car’s engine is 20 kW. The resistance to the car’s motion is 400 N.

i) Find the speed of the car at an instant when its acceleration is

ii) Show that the maximum possible speed of the car is

The work done by the car’s engine as the car travels from a point A to a point B is 1500 kJ.

Given that the car is travelling at its maximum possible speed between A and B, find the time taken to travel from A to B.

i)

ii)

iii)