Your car’s speedometer is geared to accurately
give your speed using a certain size tire. Suppose
your car has d = 14 inch diameter wheels and the
height of the tire is h = 4.5 inches.
(a) You buy a new set of tires with h = 5.5 inches and d = 14 inches. On a trip to Spokane, you maintain a constant speed of 65 mph, according to your speedometer. However, as luck would have it, you are stopped for speeding. Explain how this could happen. What did the radar gun display as your true speed?
(b) You are furious over the speeding ticket and
return to the tire dealer, demanding new tires
which are the correct size. The dealer only has “low profile tires” in stock, which are h = 3.75 inches high. If you accept these and drive away from the dealer with your speedometer reading 35 mph, how fast are you really going?
(a) Your car's speedometer is calibrated based on the size of the tires it is originally equipped with. It measures the speed by counting the revolutions of the wheels and assumes that the circumference of the tires remains constant.
When you replace the original tires with new ones that have a larger height (from 4.5 inches to 5.5 inches), the circumference of the tires increases. This means that for every revolution of the wheels, the car will travel a greater distance.
Since the speedometer is calibrated based on the original tire size, it will not accurately reflect the increased distance traveled with the new tires. As a result, the speedometer will show a higher speed than your actual speed. In other words, the speedometer will read 65 mph, but your true speed will be lower than that.
To determine your true speed, you need to calculate the ratio of the circumferences of the old and new tires. The circumference of a tire is calculated using the formula C = πd, where C is the circumference and d is the diameter.
For the old tires: C_old = π × 14 inches = 43.9823 inches
For the new tires: C_new = π × 14 inches = 43.9823 inches
The ratio of the circumferences is: C_new / C_old = 43.9823 inches / 43.9823 inches = 1
Since the ratio is 1, your true speed is the same as the speed displayed on the speedometer, which is 65 mph.
(b) If you accept the "low profile tires" with a height of 3.75 inches and drive away with the speedometer reading 35 mph, you need to determine your true speed.
Similar to the previous calculation, you need to calculate the ratio of the circumferences of the original tires and the new low profile tires.
For the original tires: C_old = π × 14 inches = 43.9823 inches
For the low profile tires: C_new = π × 14 inches = 43.9823 inches
The ratio of the circumferences is: C_new / C_old = 43.9823 inches / 43.9823 inches = 1
Since the ratio is 1, your true speed will also be 35 mph.