To understand how projectiles move, we can use math, especially something called derivatives. Let’s break this down into simpler parts.

When a projectile moves, we can describe its position, which tells us where it is at any time. The position can be written like this:

[ s(t) = -16t^2 + v_0t + s_0 ]

In this equation:

• ( v_0 ) is how fast the projectile starts moving (initial velocity).
• ( s_0 ) is where it starts from (initial height).

Now, let’s look at two important parts of motion: velocity and acceleration.

1. Velocity: The velocity tells us how fast the projectile is going. We get it from the position equation. The velocity function is:

   [ v(t) = s'(t) = -32t + v_0 ]

   This means that at any time ( t ), we can find out the speed of the projectile.

2. Acceleration: Acceleration tells us how quickly the velocity changes. We can find it by taking the derivative of the velocity function:

   [ a(t) = v'(t) = -32 ]

   This value is constant, which means the projectile is always being pulled down by gravity.

By looking at these two parts, we can learn a lot about the projectile’s journey. We can figure out when it reaches the highest point, how long it stays in the air, and how high it goes.

For example, when we set ( v(t) = 0 ), we can find the exact moment when the projectile is at its peak height.

In short, using the position, velocity, and acceleration functions helps us understand the exciting movement of projectiles!