To understand how an object falls freely, we need to look at two main ideas: motion and the laws of motion that Sir Isaac Newton gave us.

When we say "free fall," we mean that an object is moving only because of gravity. Because of this, we can make our math simpler, focusing only on the pull of gravity.

First, let’s talk about gravity.

On Earth, gravity pulls objects down at about $9.81$ meters per second squared ($m/s^2$). To make things easier in class, we often round this to $10 \, m/s^2$. It’s important to remember that this downward pull is what makes things fall toward the Earth.

Now, we can use some basic math equations about motion.

There are four main equations we use when an object is moving with constant acceleration:

1. The first equation connects how fast something is going at the start ($v_0$), how fast it’s going at the end ($v$), how quickly it speeds up ($a$), and the time it takes ($t$): $v = v_0 + at$

2. The second equation helps us find the distance ($s$) an object travels. It also includes the starting speed, time, and acceleration: $s = v_0 t + \frac{1}{2} a t^2$

3. The third equation links starting and ending speeds with distance: $v^2 = v_0^2 + 2a s$

4. The fourth equation is useful when we don’t care about time directly: $s = \frac{(v + v_0)}{2} t$

For free fall, we replace $a$ with $-g$.

The negative sign shows us that the motion is going down. We also need to consider how fast the object starts off based on the situation.

Let’s look at two different cases:

Case 1: An object is dropped from a stop.

When we drop an object without giving it any push ($v_0 = 0$), the equations change to:

• Final Speed: $v = 0 - gt \quad \Rightarrow \quad v = -gt$

• Distance Fallen: $s = 0 \cdot t + \frac{1}{2} (-g) t^2 \quad \Rightarrow \quad s = -\frac{1}{2} g t^2$

Case 2: An object is thrown down with a starting speed $v_0$.

If we give the object a push downward first, then its equations will look like this:

• Final Speed: $v = v_0 - gt$

• Distance Fallen: $s = v_0 t - \frac{1}{2} g t^2$

These examples show how changing the starting speed can change the equations of motion for falling objects.

As we look deeper into these equations, here’s what some of the letters mean:

• $t$ is the time the object has been falling.
• $s$ is how far it has gone down.

Also, the negative sign in $s$ shows that we are measuring downward, starting from where the object was dropped.

The negative speed tells us that the object is going down, reinforcing the idea that it’s speeding up as it falls.

By using these equations, we can make sense of how things fall under the influence of gravity.

To wrap it up, the equations we’ve talked about help us understand how objects in free fall behave. They show us how gravity affects motion, allowing us to predict how things will move. This knowledge is also helpful when we start considering more complex situations, like when air pushes back against objects or when objects move in different directions.

Overall, the basics we went over are the building blocks for understanding how falling works.