How do we know that 6 seconds of uniform acceleration

at 60 mph is equivalent to 6 seconds travel at 30 mph?

To understand why 6 seconds of uniform acceleration at 60 mph is equivalent to 6 seconds of travel at 30 mph, we need to consider the concept of average speed.

Average speed is defined as the total distance traveled divided by the total time taken. In this case, we have two scenarios:

1. 6 seconds of uniform acceleration at 60 mph:

To calculate the distance traveled in this scenario, we need to convert 60 mph to feet per second, as the unit of time in this case is seconds.

1 mile = 5280 feet and 1 hour = 3600 seconds, so 60 mph = (60 * 5280) / 3600 = 88 feet per second.

Given that the acceleration is uniform, we can use the formula:

distance = (initial velocity * time) + (0.5 * acceleration * time^2)

In this case, the initial velocity is 0 (starting from rest), acceleration is 88 feet per second squared, and time is 6 seconds.

distance = (0 * 6) + (0.5 * 88 * 6^2) = (0 * 6) + (0.5 * 88 * 36) = 0 + 1584 = 1584 feet.

2. 6 seconds of travel at 30 mph:

To calculate the distance traveled in this scenario, we also need to convert 30 mph to feet per second.

30 mph = (30 * 5280) / 3600 = 44 feet per second.

Since the speed is constant in this scenario, we can simply multiply the velocity by time:

distance = (velocity * time) = (44 * 6) = 264 feet.

From these calculations, we can see that while the acceleration scenario covers a greater distance (1584 feet) compared to the constant speed scenario (264 feet), both cover the same distance in the given 6-second time frame. Therefore, we can conclude that 6 seconds of uniform acceleration at 60 mph is indeed equivalent to 6 seconds of travel at 30 mph.

Acceleration is defined in terms of changes in velocity. You can have equal acceleration values at any velocity.

The acceleration produced by a given force upon a given mass DOES depend slightly upon velocity, according to the special theory of relativity. However, at the low speeds that you have mentioned, the difference is negligible.