Calculating gravity from height and time of a jump

[Icky Fizz]

Homework Statement

I'm trying to calculate a value for gravity based on a known jump height and time. It's working when the take-off and landing heights are the same but I'm struggling for a solution if the landing point is higher than the take-off. I'm more interested in understanding a solution method than getting an answer to a specific question, so I've used symbols instead of values, thanks.

An astronaut lands on a planet and jumps up to a height of 'S' and lands 'T' seconds later on a rock at height 'R'. What is the value of the planet's gravity?

Homework Equations

s = ut + 0.5*at²
v = u + at

The Attempt at a Solution

When take-off and landing are the same, I use s = ut + 0.5*at² for the falling time (half total time), with u=0, rearranging s = 0.5*at² to give an answer a = 2s / (t/2)² I think this method is fine.

With a height difference, I guess a solution is based on the same equation, but separating the jump into two parts - the initial upward part to reach the rock height on the way up, and then treating the the rest of the jump as before. But I don't seem able to figure out how much time is spent in each part of the jump?

 

[Delphi51]

Welcome to PF, Fizz!
Interesting problem! I wrote Vi for the initial upward velocity and t for the time to maximum height so I have 3 unknowns including g. That means 3 equations needed. I'm an old high school teacher so I usually only remember two accelerated motion formulas: d = Vi*t + .5*a*t and V = Vi + a*t.
I used the d one twice for distance R and distance T. And the V one for the maximum height when V = 0. It should be just a matter of eliminating t and Vi to get an answer for g. Be careful about the signs; I took acceleration to be -g so g comes out positive, but you could do it with g negative if you prefer.

 

[Icky Fizz]

Thanks for your suggestion, sounds much simpler than the quadratic equation I ended up with working out how much time was spent going up and how much coming down!