Motion in a Gravitational Field

We know from Newton's Law of Universal Gravitation that an object in the Earth's gravitational field experiences a force pulling it towards the centre of the Earth. If this is the only force acting on the object then the object will accelerate towards the centre of the Earth.

A person standing on the Earth's surface will interpret this acceleration as objects always falling downwards.

Recall that the value of the acceleration due to gravity can be treated as a having a constant magnitude of $g=\text{9.8}\text{ m·s$^{-2}$}$.

Important:

In reality, if you go very far away from the Earth's surface, the magnitude of $\vec{g}$ would change, but, for everyday problems, we can safely treat it as constant. We also ignore any effects that air resistance (drag) might have.

For the rest of this tutorial, we will deal only with the case where the force due to gravity is the only force acting on the projectile that is falling. Any projectile can be described as falling, even if its motion is upwards initially.

The initial velocity, $\vec{v}_i$, that an object has and the acceleration that it experiences are two different quantities. It is very important to remember that the gravitational acceleration is always towards the centre of the Earth and constant, regardless of the direction or magnitude of the velocity.

Tip:

Projectiles moving upwards or downwards in the Earth's gravitational field always accelerate downwards with a constant acceleration $\vec{g}$. Note: non-zero acceleration means that the velocity is changing.

Objects moving upwards or downwards in the Earth's gravitational field always accelerate towards the centre of the Earth. This looks like a downwards acceleration to someone standing on the Earth's surface. We don't draw both vectors on the object because we would be mixing velocity and acceleration which are two different physical quantities.

This means that if an object is moving upwards with some initial velocity in the vertical direction, the magnitude of the velocity in the vertical direction decreases until it stops ($v=0$ $\text{m·s $^{-1}$}$) for an instant. The point at which the velocity is reduced to zero corresponds with the maximum height, $h_{max}$, that the object reaches. After this, the object starts to fall. It is very important to remember that the acceleration is constant but the velocity vector has changed in magnitude and direction. At the maximum height where the velocity is zero the acceleration is still $\vec{g}$.

Tip:

Projectiles that have an initial velocity upwards will have zero velocity at their greatest height, $h_{max}$. The acceleration is still $\vec{g}$.

(a) An object is thrown upwards from height ${h}_{i}$. After time ${t}_{m}$, the object reaches its maximum height, and starts to drop downwards. After a time $2{t}_{m}$ the object returns to height ${h}_{i}$.

Consider an object thrown upwards from a vertical height ${h}_{i}$. We have said that the object will travel upwards with decreasing vertical velocity until it stops, and then starts moving vertically downwards. The time that it takes for the object to fall back down to height ${h}_{i}$ is the same as the time taken for the object to reach its maximum height from height ${h}_{i}$. This is known as time symmetry. This is a consquence of the uniform acceleration the projectile experiences.

This has two implications for projectiles that pass through a point on both the upward and downward part of their motion when in free-fall:

time symmetry: the time intervals during the upward motion and the downward motion are the same, for example it will take the same time to rise from initial position to maximum height as it will to drop back to the initial position. This applies to any point as shown by the pink dot in the figure above. The time interval between the projectile passing the point and being at maximum height is the same, $\Delta t$.
magnitude of velocity: the magnitude of the velocity at the same point on the upward and downward motion will be the same, the direction will be reversed, as indicated by the shaded region in the figure above.

This is very useful when solving problems because if you have any information about the upward motion you can learn something about the downward motion as well and vice versa.

Motion in a Gravitational Field