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## Acceleration due to gravity

Every falling object has the same acceleration due to gravity independent of its mass. In the ancient times it was assumed that heavier objects fall faster than lighter objects, at least until Galileo’s brilliant mind found such beliefs to be wrong.

The biggest problem with rationalising motion of free falling objects is the issue of air resistance. Air resistance slows down less dense objects more significantly than more dense objects. That is because a less dense object has a larger surface area relative to its mass and the greater the effective surface area, the greater the air resistance.

It is common knowledge that in air, a coin falls faster than a feather. Though it may seem surprising, in a vacuum a coin and a feather fall at the same rate. The difference between the two situations is that, in air the air resistance has a greater effect on a less dense body (feather) than on a more dense body (coin). The air resistance to the feather is great when compared to its weight but with the coin, the air resistance is negligible compared to its weight at low speeds.

### Value of acceleration due to gravity

All free falling bodies under the effect of gravity accelerate uniformly towards earth, provided that the air resistance is negligible. This acceleration is called the acceleration of free fall or acceleration due to gravity and is denoted by g.

The value of g varies slightly from place to place over the Earth but is constant for a chosen place. Its approximate value is 9.81m/s2 or 9.81ms-2 though it is common rounded to 10m/s2 for ease of calculation. In some situations you will find it approximated to π2 (which is about 9.87).

What all this basically means is that a free falling body increases its velocity by g every second if air resistance is negligible. The table below shows the expected changes in velocity of an object dropped from a certain height.

time (s) velocity (m/s)
0 0
1 9.81
2 19.62
3 21.43

etc

### Effect of g on rising objects

We said a free falling object accelerates at gm/s2 towards earth. Acceleration due to gravity also affect rising objects (eg objects thrown vertically upwards). A rising object decelerates at gm/s2 which basically means it is accelerating at –gm/s2. This is because it is moving against gravity.

The value of g is positive for falling objects and negative for rising objects.

## Calculations and formulae

Motion against gravity can be solved using the equations of motion because it is part of kinematics. However there are a few things we need to take into consideration:

• if an object is released from a height, its initial velocity (u) is 0m/s and its acceleration is g. Using the formulae for final velocity (v = u+at), we get:

v = gt

• if a object is thrown vertically upwards, its initial velocity (u) is the velocity of projection and its acceleration is –g. Using the formulae v = u+at, we get:

v = u-gt

• since the initial velocity of an object released from a height is 0m/s, its displacement after time t becomes:

$latex s = \frac{1}{2}gt^2$ from the equation of displacement

## Air resistance and terminal velocity

For a free falling object, air resistance opposing its motion increases as its speed increases. The air resistance reduces the acceleration of the object because its an opposing force. The air resistance increases as the square of the velocity.

Eventually, the air resistance (acting upwards) equals the weight of the object (acting downwards). When this happens, the resultant force acting on the object becomes zero since the gravitational force (weight) balances the frictional force (air resistance).

The resultant force on the object being zero, the object then falls at a constant velocity, called its terminal velocity. The value of the terminal velocity of an object whose depends on its size, shape and weight.

A small dense object, such as a coin, has a high terminal velocity and falls a large distance with an acceleration of 9.81 m/s2 before air resistance equals its weight. A less dense object, such as a feather, or an object with a large surface area relative to its weight, such as a parachute, has a low terminal velocity and only falls a short distance before air resistance equals its weight.