# Free-fall in flat-earth gravity

#### Prerequisites

One of the nice things about the flat-earth gravity approximation is that the gravitational force (the weight) is a constant and always points in the same direction. This means that when an object has nothing touching it, there is only one force acting on it — gravity (always assuming we can neglect resistive forces coming from moving through air or water) — and it is a constant. This means that Newton's laws of motion become fairly simple. But since the object can move sideways as well as up and down, there is an interesting interplay between the two directions.

If the only significant force acting on an object is gravity we say it is * in free-fall*. Be careful! This doesn't mean that it is "just falling". The object can have any initial velocity. An object traveling straight up or sideways is still in free-fall — since the only force acting on it is down. How that down force changes the "not-down" velocity is one of the interesting issues we can consider. Since objects with initial velocity are "projected", we often refer to motion in free fall as

*projectile motion.*

This is one of those "simplest possible" situations that illustrates something important — here, the vector nature of Newton's second law. As is usual in physics, when we find a "simplest possible problem" of a particular kind, we tend to beat it to death in order to learn to understand the basics before we build up to more complex systems. It's also of great importance to people who want to shoot things, like engineers, the military, and rocket scientists. Although there are some interesting examples of projectile motion in biology (such as plants shooting their seeds or jumping ants), it's not basic to biology. But since the vector character of motion and force is basic, we'll go through the fundamental ideas and a few examples.

## Vector equations for free-fall in flat earth gravity

Flat earth gravity is a place where the vector character of Newton's second law has an interesting result. Remember that Newton's second law is a *vector* law:

$$\overrightarrow{a}_A= \frac{\overrightarrow{F}^{net}_A}{m_A}$$

which means that it stands for two (in 2D, or three in 3D) equations:

$$v_x=\frac{dx}{dt}\;\;\;\;\;a_x=\frac{F_x}{m}$$

$$v_y=\frac{dy}{dt}\;\;\;\;\;a_y=\frac{F_y}{m}$$

When we have gravity, it's often (not always!) convenient to choose our x-axis to be horizontal and our y-axis to be vertical and upward. That way, the gravitational force in flat-earth gravity becomes

$$\overrightarrow{W}_{E \rightarrow A} = m \overrightarrow{g} $$

$$\overrightarrow{g} = -g \hat{j}$$

Notice! We use the convention that "g" without an arrow on it means the magnitude of g and is a positive quantity. Some introductory physics materials hide a negative sign inside the non-vector g. We feel that this has a lot of negative effects (pun intended):

- It violates our standard notation that a symbol with an arrow on it represents a vector and the symbol with that vector taken off represents the magnitude of that vector (a positive quantity).
- It assumes that we will choose the positive direction of the y-axis upward. This restricts our freedom to choose other more convenient coordinates -- say the y-axis at an angle to the vertical when we are analyzing a ramp, or pointing with its positive direction downward when we are throwing objects downward.
- By hiding the negative sign inside a symbol, it tempts us to forget that it's there and then put it in twice in different parts of a problem.

As a result, we STRONGLY recommend writing "g" as a positive quantity and putting in the negative sign if needed for the particular choice of coordinate.

## Projectile motion

All that generality aside, for describing projectile motion in free-fall, it IS convenient to choose the y axis pointing up. With this choice, and putting in that there is only one force and it points down, Newton's second law becomes the equations:

$$v_x=\frac{dx}{dt}\;\;\;\;\;a_x= 0$$

$$v_y=\frac{dy}{dt}\;\;\;\;\;a_y= -g$$

This shows that the x-motion has no acceleration so the object maintains whatever x-velocity it had when it was released. The y-motion, however, has a constant acceleration downward. This means that the y-velocity is continuously changing. There's lot's you can do with this, especially since we can use the average velocity equations in both cases, since for the x-motion we know the average velocity = instantaneous (constant) velocity, and for the y-motion we know the average velocity = (initial velocity + final velocity)/2. There are lots of relations that can be generated among the different parameters describing the situation and lots of different problems that can be solved. As an example, check out:

While these results look straightforward, they lead to a LOT of really surprising (but correct) results. Test your understanding with the follow-on example and some of the additional problems at the bottom of the page.

- Shoot and drop
- Projectile graphs
- The cut pendulum
- Free-fall acceleration

Joe Redish 10/2/11

#### Follow-ons

Last Modified: July 12, 2019