Question

Question: Suppose a car’s standard tires are replaced with tires 1.30 times larger in diameter. (i) Will the car’s speedometer reading be (a) 1.69 times too high, (b) 1.30 times too high, (c) accurate, (d) 1.30 times too low, (e) 1.69 times too low, or (f) inaccurate by an unpredictable factor? (ii) Will the car’s fuel economy in miles per gallon or km/L appear to be (a) 1.69 times better, (b) 1.30 times better, (c) essentially the same, (d) 1.30 times worse, or (e) 1.69 times worse?

Short answer

Answer

1. The speedometer reading of the car will be times low. Hence, option (d) is correct.
2. The car’s fuel economy in miles per gallon or km/L appears to be times worse. Hence, option (d) is correct.

Step-by-step solution

Defining speedometer and odometer

Speedometer is an instrument that measures the speed of moving body (generally automobile). It displays the instantaneous speed of the vehicle.

Odometer is an instrument which is used to measure the distance travelled by the vehicle.

Analysing the car’s speedometer reading

The speedometer measures the number of revolutions per second of the tire. Hence the larger tire will rotate fewer times to cover the same distance. The speedometer reading is assumed to be proportional to the rotation rate of the tires i, e $\omega =\frac{v}{R}$for a standard tire of radius R. But the actual reading is $\omega =\frac{v}{1.3R}$or times smaller. Hence option(d) is the correct answer.

Analysing the car’s odometer reading

If we use the odometer reading for calculating the fuel economy, the reading will be a factor of , since odometer assumes $1 \text{rev}=2\pi R$ where ‘R’ is the tire radius. The actual distance travelled is $1.3\left(2\pi R\right)$. So the fuel economy in miles per gallon appears too small by a factor of . Thus, option (d) is the correct answer.