Heavier objects experience a greater gravitational force, but they don't fall together. Why?
Near Earth’s surface, the gravitational force on an object has magnitude mg. A greater mass experiences a greater gravitational force, but the force needed for the same acceleration is also proportional to mass. Substituting F = mg into Newton’s law F = ma gives a = g: mass cancels.
Therefore, if two objects are placed simultaneously at the same height and come to rest at the same time, they will arrive at the ground together, assuming there is no air resistance. This conclusion includes the condition that the point of release and the initial velocity are the same. It is not possible to compare an object that has been thrown upwards with an object that has been simply placed, under the same conditions.
How will the speed and the falling distance change depending on time?
Assuming a positive downward direction, when placed in a stationary state, the velocity is v = gt. The distance fallen is h = ½gt². The velocity is proportional to time, but the distance fallen is proportional to the square of time. The distance fallen during 1 seconds is greater than the distance fallen during 1 seconds.
Fall time is t = √(2h/g). Doubling the height multiplies the time by √2, rather than 2. On the Moon, where g is smaller than on Earth, falling from the same height takes longer. Here g is a positive magnitude; if upward is positive, the acceleration carries a minus sign.
What are the differences between a real fall and a theoretical model?
In air, resistance depends on an object’s shape, cross-sectional area, and speed. This is why a flat sheet of paper and the same sheet crumpled up can take different times to fall despite having the same mass. When air resistance balances gravity, acceleration becomes 0 and the object can approach terminal velocity.
This page’s formula assumes a constant g, negligible air resistance, and a stationary starting point. However, because ‘g’ varies depending on location in very high places or far from celestial bodies, it cannot be applied directly.
Key relationships and symbols
v = gt · h = ½gt² · t = √(2h/g)
| Symbol | Meaning | Unit |
|---|---|---|
| g | Magnitude of gravitational acceleration | m/s² |
| h | Downward distance from the starting point | m |
| t | Fall time | s |
| v | Magnitude of downward velocity | m/s |
Worked example
How long will it take for a ball, dropped from a height of 19.6 m, to reach the ground?
If g = 9.8 m/s² then t = √(2 × 19.6 / 9.8) = 2 s. The speed just before arrival is 19.6 m/s. Changing only the mass results in these two values remaining the same.
Common misconceptions
- The mass independence of ideal free fall does not mean that all objects always fall together in air.
- The velocity is not constant during descent, but rather acceleration is constant.