Question

If a car generates 18 hp when traveling at a steady what must be the average force exerted on the car due to friction and air resistance?

Short answer

The average force exerted on the car due to friction and air resistance is 508.64 N.

Step-by-step solution

Identification of the given data

The power generated by the car engine is\(P = 18\;{\rm{hp}} \times \frac{{746\;{\rm{W}}}}{{1\;{\rm{hp}}}} = 13428\;{\rm{W}}\).

The steady speed of the car is \(v = 95\;{\rm{km/h}} \times \frac{{1\;{\rm{m/s}}}}{{3.6\;{\rm{km/h}}}} = 26.4\;{\rm{m/s}}\).

Definition of power

Power, P is defined as the rate at which work is done and is obtained by dividing the time taken to perform that work. It is given as:

\(P = \frac{W}{t}\)... (i)

Relationship between power and force

The average power in terms of force can be written as:

\(\begin{aligned}{P_{{\rm{av}}}} &= \frac{{{\rm{Work}}}}{{{\rm{time}}}}\\ &= \frac{{{\rm{Force}} \times {\rm{distance}}}}{{{\rm{time}}}}\\ &= {\rm{Force}} \times {\rm{speed}}\\ &= Fv\end{aligned}\) … (ii)

Here, the force and speed are in the same direction.

Determination of average force exerted by the car

The power generated by the engine creates a force on the ground to propel the car in the forward direction.

Thus, the force required to propel the car in the forward direction is given by equation (ii). It is given that the car moves with a steady velocity.The resistive force is of the same magnitude as the engine force, and so the net force is zero.

Thus, the average force exerted on the car due to friction and air resistance is given by equation (ii).

\(\begin{aligned}F &= \frac{{{P_{{\rm{av}}}}}}{v}\\ &= \frac{{13428\;{\rm{W}}}}{{26.4\;{\rm{m/s}}}}\\ &= 508.64\;{\rm{N}}\end{aligned}\)

Thus, the resistive force is 508.64 N.