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Power
A vehicle has mass m = 800kg  
The total force (due to friction, air resistance etc) opposing its motion, when moving at 20ms-1 is FR = 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 FR 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 FM = 1142  
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!)   
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