Need help with a tough physics question? calculating gravitational acceleration?

Question

You land on an unknown planet somewhere in the universe that clearly has weaker gravity than Earth. To measure g on this planet you do the following experiment: A ball is thrown upward from the ground. It passes a windowsill 12.0 m above ground and is seen to pass by the same windowsill 2.00 s after it went by on its way up. It reaches the ground again 5.00 s after it was thrown. Calculate the magnitude of g (the acceleration due to gravity) at the surface of this planet.

I have already tried using the position kinematic, using time 2.5 and time 5. re substituting back Vi into the first equation and solving for g i got 0.8333

Not sure what i'm doing wrong

Answer 1

Use this function for position x at time t:

x(t) = x0 + (v0)t - ½gt²

x0 is the initial height, which is zero, so:

x(t) = (v0)t - ½gt²

At some unknown time t1, it passes the 12m windowsill:

x(t1) = 12.0 = (v0)(t1) - ½g(t1)²

It's also at 12m two seconds later:

x(t1+2.0) = 12.0 = (v0)(t1+2.0) - ½g(t1+2.0)²

Finally, it's back on the ground (x=0) after 5 seconds:

x(5) = 0 = (v0)(5) = ½g(5)²

Recapping the three equations:

12.0 = (v0)(t1) - ½g(t1)²

12.0 = (v0)(t1+2.0) - ½g(t1+2.0)²

0 = (v0)(5) - ½g(5)²

These are 3 equations in 3 unknowns (t1, v0 and g), so you should be able to find unique solutions for all the variables, using ordinary algebra.