A vehicle has mass m = 800kg 

The total force (due to friction, air resistance etc) opposing its motion,
when moving at 20ms^{1} is F_{R}
= 750N.


Calculate the power needed from the motor in order to maintain a steady
speed of 20ms^{1} while driving up
a 5% hill. 

(A 5% hill is one for which the angle of the slope is such that
its tangent is 0.05.) 



To maintain a steady speed up the hill, the force produced by the motor must be
equal but opposite to the sum of the force of resistance to motion F_{R}
and the component of the force of gravity acting down the slope (the
red arrow in the diagram). 

Considering magnitudes only we have 





Cosθ is equal to the sin of the angle of the slope,
x. 





The angle of the slope, x was stated to be tan^{1}
0.05. 

This is a small angle (about 2.86°) 

For such a small angle, the difference between the
lengths a and h (in the right angled triangle drawn here to represent the slope)
is very small. 

For this reason, the sin and tan are virtually identical so we can
simply replace cosθ
with 0.05 to a very good approximation. 

This gives F_{M} = 1142N 

Now, we simply use 



so, in this case, the power required from the motor is 



Power = 22.84kW 

(which might sound a bit high for a "2cv" but... it's only an
illustration of the principle!) 
