The amount of gasoline needed to travel a given distance at a constant speed is

Question

The amount of gasoline needed to travel a given distance at a constant speed is determined by the distance and the dissipative forces that must be overcome. For a vehicle, there are two distinct dissipative forces: (1) the friction-like force (called "rolling resistance"), which is proportional to the weight of the car and is largely independent of speed, and (2) the force of air resistance, which is proportional to the square of the speed. Suppose your car has a mass m = 2500 kg, and friction is given by F_fric = 0.018 mg, while the force of air resistance is F_air = (0.6 kg/m) v^2. (1) Compute how much work must be done against the sum of these forces if you travel 100 km (a) at a speed v = 15 ms/(~34 mph), and (b) if you travel the same distance at v = 30 m/s (~67 mph).

Explanation / Answer

1) v = 15 m/s

F = 0.018 x 2500 x 9.81 + 0.6 x 152

F = 576.45 N

work = F.ds = 576.45 x 100000 = 57.645 x 106 J = 57.645 MJ

when v = 30 m/s

F = 0.018 x 2500 x 9.81 + 0.6 x 302

F = 981.45 N

work = F.ds = 981.45 x 100000 = 98.145 x 106 J = 98.145 MJ

2) a) no. of gallons used = 57.645/(121/4) = 1.90562 gallons

b) no. of gallons used = 98.145/(121/4) = 3.24446 gallons

3) a) mileage = 100/1.90562 km/gallon = 52.476364 km/gallon = 32.6073 mile /gallon

b) mileage = 100/3.24446 km/gallon = 30.8217 km/gallon = 19.15174 mile/gallon

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