How can you prove that acceleration due to gravity is the same in a resting and moving frame?
Cristina, I have no idea what level you are, so I am going to assume you are not dealing with Lorenez factors.
But let me summarize on the force equation.
F=ma=m v/t' where t is the dilated time at relativistic speeds. Under constant acceleration, mass does not change, but because of the high speeds, time is dilated, so it takes a greater force to accelerate it. But with gravity, the greater force is provided by the dilated mass already present, so in fact, under the force of gravity, with dilation of time, the acceleration remains the same.
For a detailed argument,
http://www.mrelativity.net/TimeEnergyIG/TimeEnergyIG2.htm
And the only way to actually prove it is to test it, which has been done.
The apparent acceleration of a body that is only subjected to gravity is independent of the velocity of the coordinate system, as long that system is moving in a straight line at constant velocity. If the coordinate system is accelerating, there will be other apparent forces (Coriolis and pseudo-gravity) that lead to additional acceleration in that coordinate system. In a linearly accelerating coordinate system, there is no way to tell if the additional apparent force is due to gravity or the motion of the coordinate system. (A related statement is that there is no way to tell if you are really weightless in a free-falling elevator). This is one of the imporant principles that led Einstein to the General Theory of Relativity.
The short answer to your question is that the acceleration due to gravity alone IS independent of coordinate system motion, but the proof is complex and reduces to the fact that that experiment supports it.
To prove that acceleration due to gravity is the same in a resting and moving frame, you can follow these steps:
1. Understand the concept: Acceleration due to gravity refers to the acceleration an object experiences due to the gravitational force exerted by a massive body, such as the Earth. Its value is approximately 9.8 m/s² near the Earth's surface.
2. Frame of reference: A frame of reference is a coordinate system used to describe the motion of objects. In this case, we have two frames of reference: one at rest (stationary), and one in motion (moving at a constant velocity).
3. Equivalence principle: The Equivalence Principle states that the effects of gravity are indistinguishable from the effects of acceleration. This means that in an accelerating frame of reference, objects will experience the same effects as if they were in a gravitational field.
4. Thought experiment: Imagine you are inside an elevator. In the resting frame (stationary), you feel your weight as the force of gravity pulls you downward. Now, imagine the elevator starts moving at a constant velocity. In this case, you are subject to both the force of gravity and the backward force from the elevator’s acceleration. While the elevator moves, you should notice that your weight remains unchanged.
5. Experimental verification: The famous experiments conducted by Galileo Galilei and later by astronaut David Scott on the Moon provide direct evidence of the equivalence between gravity and acceleration. Galileo performed experiments by dropping objects from the Leaning Tower of Pisa, and David Scott dropped a feather and a hammer on the Moon during the Apollo 15 mission. In both cases, the objects fell at the same rate, demonstrating that the acceleration due to gravity is independent of the frame of reference.
By considering the equivalence principle, thought experiments, and actual experiments, one can conclude that the acceleration due to gravity remains the same in both a resting and moving frame of reference.