**Projectile Motion III: Varying gravity**

We can also do some interesting things with projectile motion if we vary the gravitational pull when we look at projectile motion. The following graphs are all plotted in parametric form.

Here t is the parameter, v is the initial velocity which we will keep constant, theta is the angle of launch which we will vary, and g is the gravitational constant which on Earth we will take as 9.81 m/s^{2}.

**Earth **

Say we take a projectile and launch it with a velocity of 10 m/s. When we vary the angle of launch we get the folowing graphs:

On the y axis we have the vertical height, and on the x axis the horizontal distance. Therefore we can see that the maximum height that we achieve is around 5m and the maximum horizontal distance is around 10m.

**Other planets and universal objects**

We have the following values for the gravitational pull of various objects:

Enceladus (Moon of Saturn): 0.111 m/s^{2}, The Moon: 1.62 m/s^{2}, Jupiter: 24.8 m/s^{2}, The Sun: 274 m/s^{2}, White dwarf black hole surface gravity: 7×10^{12}m/s^{2}.

So for each one let’s see what would happen if we launched a projectile with a velocity of 10 m/s. Note that the mass of the projectile is not relevant (though it would require more force to achieve the required velocity).

**Enceladus:**

**The Moon:**

**Jupiter:**

**The Sun:**

**Black hole surface gravity:**

This causes some issues graphically! I’ll use the equations derived in the last post to find the coordinates of the maximum point for a given launch angle theta:

Here we have v = 10 and g = 7×10^{12}m/s^{2}. For example if we take our launch angle (theta) as 45 degrees this will give the coordinates of the maximum point at:

(7.14×10^{-12}, 3.57×10^{-12}).

**Summary:**

We can see how dramatically life would be on each surface! Whilst on Earth you may be able to throw to a height of around 5m with a launch velocity of 10 m/s., Enceladus would see you achieve an incredible 450m. If you were on the surface of the Sun then probably the least of your worries would be how hight to throw an object, nevertheless you’d be struggling to throw it 20cm high. And as for the gravity at the surface of a black hole you wouldn’t get anywhere close to throwing it a nanometer high (1 billionth of a meter).

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