One way to measure g on another planet or moon by remote sensing is to measure how long it takes an object to fall a given distance. A lander vehicle on a distant planet records the fact that it takes 3.17 s for a ball to fall freely 11.26 m, starting from rest. What is the acceleration due to gravity on that planet?
x = 1/2 a t^2 ==> a = (2x) / t^2
a = 2(11.26m) / (3.17s)^2
a = 22.52m / 10.0489s^2 = 2.241041308 ==> sig figs
a = 2.24m/s^2
To find the acceleration due to gravity on the distant planet, we can use the equation of motion for freefall:
s = ut + (1/2)gt^2
Where:
s = distance fallen (11.26 m)
u = initial velocity (0 m/s, because the ball starts from rest)
t = time taken (3.17 s)
g = acceleration due to gravity (what we need to find)
Plugging in the values, we get:
11.26 = 0*(3.17) + (1/2)g*(3.17)^2
Now, we can solve for g:
11.26 = (1/2)g*(3.17)^2
Multiply both sides by 2 and divide by (3.17)^2:
g = (11.26 * 2) / (3.17)^2
g = 68.27 / 10.0889 ≈ 6.769 m/s^2
Therefore, the acceleration due to gravity on that planet is approximately 6.769 m/s^2.
To find the acceleration due to gravity on the distant planet, we can use the equation of motion for free fall:
d = (1/2) * g * t^2
where:
d is the distance (11.26 m),
g is the acceleration due to gravity (what we want to find), and
t is the time it takes for the object to fall (3.17 s).
Rearranging the equation, we have:
g = (2 * d) / t^2
Substituting the given values, we get:
g = (2 * 11.26 m) / (3.17 s)^2
Calculating this expression gives:
g ≈ 7.88 m/s^2
Therefore, the acceleration due to gravity on that distant planet is approximately 7.88 m/s^2.