Equations of Motion

Computing position, velocity, and acceleration of a falling object

Exercise 1

A coin is tossed downward from a height of 60 m with an initial velocity of 2 m/s. Assuming no other forces are acting on the coin, determine the following.

How many seconds does it take for the coin to hit the ground?

What is the coin’s instantaneous velocity (in m/s) just before hitting the ground?

What is the coin’s acceleration...

... at the moment it is thrown (t=0)?

... just before hitting the ground?

Setting up functions

Start with the equation for finding the final position (Subscript[x, f]) (in this case, the ground):

Plug in known values for (Subscript[x, i]), (Subscript[v, i]), and a to represent (Subscript[x, f]) as a function of t:

The first two derivatives represent velocity and acceleration:

Visualizing functions

Plot the position, velocity and acceleration of the coin over time:

Plot[Evaluate[{x[t], v[t], a[t]}], {t, 0, 4},PlotLegends->{"x(t)","v(t)","a(t)"}]

You can see that the path of the coin, x(t), is a parabola that opens downward. The point where x(t) crosses the horizontal axis represents the approximate time when the coin hits the ground. Since the coin is only affected by Earth’s gravity, v(t) decreases linearly and a(t) remains constant.

Solutions

How many seconds does it take for the object to hit the ground?

What is its instantaneous velocity just before hitting the ground?

What is the coin’s acceleration...

... at the moment it is thrown (t=0)?

... just before hitting the ground?

Bonus: How could you answer problem 3 without referring to a(t)? (Hint: Consider the source of the acceleration.)