How to find: instantaneous speed vs average speed without the use of calculus?

Question

Suppose you hold a ball out over a cliff and let go. how would you find the instantaneous speed after 10 seconds and the avg speed after 10 seconds?

I thought you could just find the displacement of x over the time span of 10 seconds and then divide the displacement of x by 10 seconds.

Am I wrong?

Answer 1

Hi Sofia,

You are not wrong, you definitely can find the average velocity by dividing displacement by the time or the average speed by dividing the distance traveled by the time. This universal method works for any motion. However for the motion with constant acceleration (constant acceleration can also be zero, by the way) there are useful "shortcuts", namely the kinematic equations, which are not requiring calculus.

To find the instantaneous velocity (or its magnitude, the instantaneous speed) we can use the equation

v(t) = v₀ + at

In your case,

v₀ = 0 m/s

a = 10 m/s²

t = 10 s

Therefore v = 0 m/s + (10 m/s²)(10 s) = 100 m/s

To find the average velocity (which magnitude in your case is the average speed) you can use the formula

v_average = (v₀ + v) / 2

In your example

v₀ = 0 m/s

v = 100 m/s

Therefore the average speed is

v_average = (0 m/s + 100 m/s) / 2 = 50 m/s

Now let us check our answer. In 10 seconds our object traveled the distance of

x = v₀t + 1/2 a t² = (0.5)(10 m/s²)(10s)² = 500 m

Its average speed is thus

v_average = 500 m / 10s = 50 m/s

Best,

Sofia