# Speed of falling objects / with different altitudes

I have a question about gravity

So as we know gravity is significantly less on high mountains or tall buildings and increases as we lose height

☕ My question is that if we have 2 objects with the same volume and mass, 1 falls on a mountain and one falls on a beach ( note that they are released from the same height based on the surface they shall fall on ) Will they both reach the ground at the same time or will their speeds be different?

• Can we assume you're ignoring air resistance? Commented Sep 5, 2022 at 21:01
• Gravity is not significantly less on high mountains or tall buildings. Commented Sep 5, 2022 at 23:32

## 2 Answers

Gravity on the top of Mount Everest is approximately the same as gravity at sea level, with difference lower than $$1\%$$, usually negligible if you're doing such a kind of experiment.

Anyway, the answer is yes, the ball at the sea level takes less time than the ball on the top of the mount Everest, because the gravity is stronger and because the motion is governed by the same equation

$$y(t) = h - \dfrac{1}{2} g t^2$$,

where $$g$$ is the local intensity of the gravity field (approximately constant in the domain of your experiment), so that the time to reach the ground, i.e. when $$y(t_G)=0$$, is

$$t_G = \sqrt{\dfrac{2 h}{g}}$$.

Since $$g^{sea} > g^{Eve}$$, you get

$$t^{sea}_G = \sqrt{\dfrac{2 h}{g^{sea}}} < \sqrt{\dfrac{2 h}{g^{Eve}}} = t^{Eve}_G$$.

• "The answer is yes." I think you need to clarify what you mean, since OP asked a bad/compound question (two different question separated by an "or"). The answer is "no" to the first (the will not reach the ground at the same time) and "yes" to the second (their speeds will be different).
– hft
Commented Sep 5, 2022 at 20:22
• The equation for the time to hit the ground is the same, but the value of "g" is different, so the time is different.
– hft
Commented Sep 5, 2022 at 20:23
• Thank you so much Commented Sep 5, 2022 at 20:24
• @hft I'll edit the answer, to be more clear Commented Sep 5, 2022 at 20:26
• Forgive me, I was reading your answer again when another question came up in my mind: is this answer true when there isn't any air resistance force? Commented Sep 10, 2022 at 21:20

At different points on Earth's surface, the free fall acceleration ranges from 9.764 to 9.834 $$m/s^2$$.

$$9.834 / 9.764 \approx 1.007$$

https://en.wikipedia.org/wiki/Gravitational_acceleration

$$\text{speed}= g \space \text{time}$$

$$\text{distance} = \frac{1}{2} g \space \text{time}^2$$

$$\text{g}= \text{acceleration of gravity}$$

$$g_\text{low} = 1.007 \space g_{\text{high}}$$

$$\frac{\text{speed}_\text{low}}{\text{speed}_\text{high}} = 1.007$$

$$\frac{\text{distance}_\text{low}}{\text{distance}_\text{high}} = 1.007$$

• In the OP's question, the two distances are the same. Commented Sep 5, 2022 at 21:37
• The conclusion is obvious from the fact that, in any given amount of time, the lower ball will travel a further distance. The lower ball is always ahead. Commented Sep 5, 2022 at 21:45
• Which conclusion? Commented Sep 7, 2022 at 9:15
• @PM2Ring The lower ball will win the race. Commented Sep 7, 2022 at 21:37
• Ok, so you should state that explicitly in your answer, since the OP asks "Will they both reach the ground at the same time". Commented Sep 7, 2022 at 21:42