Set 3 Problem number 3

An engine drives a vehicle a distance of 8 meters.

The vehicle carries enough fuel to supply 152 Joules of energy.
How great a constant force can theoretically be supplied by the engine before exhausting its fuel, assuming all the fuel energy could be converted to useful work (this is in fact impossible, as we will see later; but the calculation is important as a reference value in calculating efficiency)?

Solution

The work done in the 8 meter distance must be equal to the 152 Joules of energy used up.

Since work equals parallel force times displacement, 152 Joules is equal to 8 meters multiplied by the force.
This could be written as an equation

152 J = ( 8 m)(force).

Solving we find that

force = 152 J/( 8 m) = 19 Newtons.

Generalized Solution

Since work is the product of parallel force and distance, the work necessary to dissipate energy `dE over a distance `ds by a parallel force F will be

`dW = E = F `ds and
F will therefore be the quotient

F = `dW / `ds, or equivalently F = `dE / `ds.

Explanation in terms of Figure(s), Extension

The figure below shows the relationships among force, displacement and work for the case of force parallel to displacement.

The relationship F = `dW / `ds tells us, for example, that if we wish to do a lot of work `dW over a long distance `ds we must exert only a little force.
The relationship `ds = `dW / F tells us, for example, that if we wish to do of work `dW using a relatively large force we need only exert that force through a relatively short distance.
The relationship `dW = F `ds tells us, for example, that the more force we exert and the greater the distance over which we exert it, the more work we do.