Question

As an astronaut visiting the Planet X, you're assigned to measure the free-fall acceleration. Getting out your meter stick and stop watch, you time the fall of a heavy ball from several heights. Your data are as follows:Height (m) Fall Time (s)0 01 0.542 0.723 0.914 1.015 1.17a. Analyze these data to determine the free-fall acceleration on Planet X.b. Determine the uncertainty in the free-fall acceleration.

Answer 1

Answer:

$g \approx 7.4 m/s^2$

Explanation:

Assuming the following data:

Heigth (m): 0 , 1, 2, 3, 4, 5

Time (s): 0.00, 0.54,0.73, 0.91, 1.01, 1.17

We know from kinematics that the height is given by the following expression:

$h_f = h_i + v_i t + \frac{1}{2} g t^2$

Assuming for this case that the initial velocity is $v_i$ we can find a polynomial with degree 2 in order to have an estimation for the height with the time.

We can use excel for this and we can see the polynomial adjusted for the data given.

As we can see the best polynomial of degree 2 is given by:

$h(x)= -0.0178 +0.066 x+ 3.6801 x^2$

For our case x = time and we can rewrite the expression like this

$h(t)= -0.0178 +0.066 t+ 3.6801 t^2$

And if we are interested on the gravity we want on special the last term of this equation, we can set equal the following terms:

$3.6801 t^2 = \frac{1}{2} g t^2$

And solving for g we got:

$2*3.6801 = g= 7.36 \frac{m}{s^2}$

So then a good approximation for the gravity of the planet rounded to 2 significant figures is 7.4 m/s^2