I was lying in bed one night when I suddenly realised that it ought to be possible to derive the equation for the time taken for a falling object to fall using nothing more than the fact that gravity causes acceleration.

Preamble

A quick warning

This post documents my original approach which uses calculus. For those who have never studied calculus before, it might be wise to check out my other post which goes through the exact same motions but uses the equivalent geometry instead of the calculus.

I would heartily recommend learning some calculus at some point if you are interested in this kind of thing though as not all equations can be attacked geometrically.

A quick calculus refresher

Differentiation

Differentiation is the process of finding some function

which allows us to calculate the gradient (change in y over change in x) of the function

at any point x

For example, the derivative of 2x is 2, 5x is 5, x² is 2x etc

We are going to need a few differentiation rules

The derivative of x to the a, where a is some constant, is a lots of x to the (a-1)

The derivative of ax, where a is some constant, is a

Integration

Integration is the process of undoing differentiation and allows you to calculate the area (average y value times the length of x) under a graph between two points. For example the integral of 2x is x², 2 is 2x, 5 is 5x etc.

We are also going to need a few integration rules:

To integrate something we need to figure out what the function which we are integrating is the derivative of

The equivalent to the constant rule

is

as

The equivalent to the power rule

is

as

Another useful rule is that

as

The main event

Setting the scene

From the moment you drop any object gravity begins to accelerate it and won’t stop unless something pesky like air resistance gets in the way. For the purposes of easier calculation, we are going to ignore air resistance.

On earth gravity has a strength of roughly 10 m/s/s, this means that after 1 second of falling the object will be travelling at 10 m/s after 2 seconds 20m/s, 3 seconds 30m/s etc. This is what makes this problem so interesting, as time goes by the velocity of the object increases, this means that as the metres go by the object gets faster and faster. In other words, it takes the object longer to do the first metre than it does to the second and so on.

The actual main event

That was kind of a mess but essentially:

Velocity (v) is change in distance (s) over change in time (t) and therefore velocity is the derivative of distance with respect to time

Acceleration is change in velocity over change in time and therefore acceleration is the derivative of velocity with respect to time and therefore (due to bullet point number one) acceleration is the second-order derivative of distance with respect to time

2. We let the acceleration be some constant g (on earth this is approximately 9.8) and integrate with respect to t using the constant rule for integration, this changes the second-order derivative to a first-order derivative

3. We integrate again this time using rule number 3 and this time we get change in distance on its own

4. We now have an equation that allows us to find out how far an object has fallen if we now how long it has been falling for, to get the equation we are aiming for we simply need to rearrange

5. And indeed upon checking, I must stress that I worked this all out on paper prior to looking up the formula, we find that this is indeed the correct equation.

Some problems

Here are a couple of problems to have a go at (feel free to leave answers in the comments), unless stated otherwise you may assume that gravitational acceleration is 9.8 m/s/s and that air resistance is negligible:

1a) A man inadvertently drops his phone from a ledge, he immediately yells to tell his friend on a balcony 2m bellow to catch it, how long does his friend have to react?

1b) His friend fails to catch it, by the time it hits the pavement the phone has been travelling for 4 seconds, how far above the pavement does the man live?

2a) An astronaut drops a golf ball on the moon from a height of one metre, how long will it take for the golf ball to hit the ground? Take gravitational acceleration to be 1.62 m/s/s

b) The astronaut repeats the experiment but this time from 0.5m, how long will it take for the golf ball to hit the ground? Take gravitational acceleration to be 1.62 m/s/s

c) The astronaut repeats the experiment but this time from 5m, how long will it take for the golf ball to hit the ground? Take gravitational acceleration to be 1.62 m/s/s

3a) A robot on Jupiter falls into a 50m deep crater, analysis of data collected by instruments shows that it was falling for 2 seconds, what is the value for gravitational acceleration on Jupiter?

3b) Hence or otherwise, how long would the robot be falling for if it fell into a 200m crater?