exupero's blog

Jumping higher when the sun is overhead

How much higher could someone jump when the sun's gravity is pulling them up? Gravitational acceleration is expressed as a relation of mass, distance, and a gravitational constant:

When the sun is directly overhead, we calculate the net gravitational acceleration by subtracting the sun's pull from the Earth's pull:

Earth's gravity pulls someone at the (average) surface with an acceleration of 9.82 m/s². At perihelion, where the sun's gravity is strongest, having the sun directly overhead reduces the net acceleration to 9.81 m/s², about 0.001% less than normal.

To find the result in more practical terms, we can calculate a theoretical difference in jumping height using a kinematic equation with no time term:

vi is the initial velocity, vf is the final velocity, a is the constant acceleration, and d is distance. For a vertical jump, the maximum height d is reached when vf is 0:

Solving for initial velocity,

We can use this vi in the same kinematic equation but with a different value for acceleration, distance now being the unknown:

In other words, gravitational acceleration and jump height are inversely proportional. Also notice that the units of a and a₂ cancel out, meaning they don't have to be compatible with the units of d and d₂.

Thus, supposing a 1-foot vertical jump under Earth-only gravity, jump height when the sun is directly overhead is (theoretically) 1.0006 feet, or about 0.2 millimeters higher—a tiny difference, but more visible than I expected.