Screen Shot 2020-04-08 at 1.09.39 PM

This carries on our exploration of projectile motion – this time we will explore what happens if gravity is not fixed, but is instead a function of time.  (This idea was suggested by and worked through by fellow IB teachers Daniel Hwang and Ferenc Beleznay).   In our universe we have a gravitational constant – i.e gravity is not dependent on time.  If gravity changed with respect to time then the gravitational force exerted by the Sun on Earth would lessen (or increase) over time with all other factors remaining the same.

Interestingly time-dependent gravity was first explored by Dirac and some physicists have tried to incorporate time dependent gravity into cosmological models.  As yet we have no proof that gravity is not constant, but let’s imagine a university where it is dependent on time.

Projectile motion when gravity is time dependent

Screen Shot 2020-04-10 at 2.12.03 PM

We can start off with the standard parametric equations for projectile motion. Here v is the initial velocity, theta is the angle of launch, t can be a time parameter and g is the gravitational constant (9.81 on Earth).  We can see that the value for the vertical acceleration is the negative of the gravitational constant.  So the question to explore is, what if the gravitational constant was time dependent?  Another way to think about this is that gravity varies with respect to time.

Linear relationship

If we have the simplest time dependent relationship we can say that:

Screen Shot 2020-04-08 at 1.01.41 PM

where a is a constant.  If a is greater than 0 then gravity linearly increases as time increases, if a is less than 0 than gravity linearly decreases as time increases.  For matters of slight convenience I’ll define gravity (or the vertical acceleration) as -3at.  The following can then be arrived at by integration:

Screen Shot 2020-04-10 at 2.17.28 PM

This will produce the following graph when we fix v = 10, a = 2 and vary theta:

Screen Shot 2020-04-08 at 1.09.39 PM

Now we can use the same method as in our Projectile Motion Investigation II to explore whether these maximum points lie in a curve.  (You might wish to read that post first for a step by step approach to the method).

Screen Shot 2020-04-08 at 1.44.34 PM

therefore we can substitute back into our original parametric equations for x and y to get:

Screen Shot 2020-04-08 at 1.46.02 PM

We can plot this with theta as a parameter.  If we fix v = 4 and a =2 we get the following graph:

Screen Shot 2020-04-08 at 1.33.06 PM

Compare this to the graph from Projectile Motion Investigation II, where we did this with gravity constant (and with v fixed as 10):

Screen Shot 2020-04-06 at 9.34.04 PM

The Projectile Motion Investigation II formed a perfect ellipse, but this time it’s more of a kind of egg shaped elliptical curve – with a flat base.  But it’s interesting to see that even with time dependent gravity we still have a similar relationship to before!

Inverse relationship

Let’s also look at what would happen if gravity was inversely related to time.  (This is what has been explored by some physicists).

In this case we get the following results when we launch projectiles (Notice here we had to use the integration by parts trick to integrate ln(t)).  As the velocity function doesn’t exist when t = 0, we can define v and theta in this case as the velocity and theta value when t = 1.

Screen Shot 2020-04-10 at 2.34.42 PM

Now we use the same trick as earlier to find when the gradient is 0:

Screen Shot 2020-04-10 at 2.37.22 PM

Substituting this back into the parametric equations gives:

Screen Shot 2020-04-10 at 2.42.57 PM

The ratio v/a will therefore have the greatest effect on the maximum points.

v/a ratio negative and close to zero:

v = 40, a = -2000, v/a = -0.02

Screen Shot 2020-04-10 at 2.52.57 PM

This gives us close to a circle, radius v, centred at (0,a).

v = 1, a = -10, v/a = -0.1

Screen Shot 2020-04-10 at 2.59.20 PM

Here we can also see that the boundary condition for the maximum horizontal distance thrown is given by x = v(e).

v/a ratio negative and large:

v = 40, a = -2, v/a = -20.

Screen Shot 2020-04-10 at 2.48.30 PM

We can see that we get an egg shape back – but this time with a flatter bulge at the top and the point at the bottom.  Also notice how quickly the scale of the shape has increased.

v/a ratio n/a (i.e a = 0)

Screen Shot 2020-04-10 at 3.07.17 PM

Here there is no gravitational force, and so projectiles travel in linear motion – with no maximum.

Envelope of projectiles for the inverse relationship

This is just included for completeness, don’t worry if you don’t follow the maths behind this bit!

Screen Shot 2020-04-15 at 10.09.59 AM

Screen Shot 2020-04-15 at 10.10.11 AM

Therefore when we plot the parametric equations for x and y in terms of theta we get the envelope of projectile motion when we are in a universe where gravity varies inversely to time.  The following graph is generated when we take v = 300 and a = -10.  The red line is the envelope of projectiles.

Screen Shot 2020-04-15 at 10.11.19 AM

A generalized power relationship

Lastly, let’s look at what happens when we have a general power relationship i.e gravity is related to (a)tn.  Again for matters of slight convenience I’ll look at the similar relationship -0.5(n+1)(n+2)atn.

Screen Shot 2020-04-10 at 2.21.52 PM

This gives (following the same method as above:

Screen Shot 2020-04-08 at 6.48.11 PM

Screen Shot 2020-04-08 at 6.48.46 PM

As we vary n we will find the plot of the maximum points.  Let’s take the velocity as 4 and a as 2.  Then we get the following:

When n = 0:

Screen Shot 2020-04-08 at 8.18.07 PM

When n = 1:

Screen Shot 2020-04-08 at 8.16.42 PM

When n =2:

Screen Shot 2020-04-08 at 8.18.21 PM

When n = 10:

Screen Shot 2020-04-08 at 8.18.44 PM

We can see the general elliptical shape remains at the top, but we have a flattening at the bottom of the curve.

When n approaches infinity:

Screen Shot 2020-04-08 at 8.26.45 PM

We get this beautiful result when we let n tend towards infinity – now we will have all the maximum points bounded on a circle (with the radius the same as the value chosen as the initial velocity.  In the graph above we have a radius of 4 as the initial velocity is 4. Notice too we have projectiles traveling in straight lines – and then seemingly “bouncing” off the boundary!

If we want to understand this, there is only going to be a very short window (t less than 1) when the particle can upwards – when t is between 0 and 1 the effect of gravity is effectively 0 and so the particle would travel in a straight line (i.e if the initial velocity is 5 m/s it will travel 5 meters. Then as soon as t = 1, the gravity becomes crushingly heavy and the particle falls effectively vertically down.