### Basic Equations and Parabolic Path

### Projectile Motion

Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning on the trajectory, after which the only interference is from gravity. In a previous atom we discussed what the various components of an object in projectile motion are. In this atom we will discuss the basic equations that go along with them.

### Initial Velocity

The initial velocity can be expressed as x components and y components:

In this equation, u stands for initial velocity, g refers to gravity, h_{o }to initial height, and θ refers to projectile angle.

### Time of Flight

The time of flight of a projectile motion is the time from when the object is projected to the time it reaches the surface. As we discussed previously, T depends on the initial velocity and the angle of the projectile, θ:

### Acceleration

In projectile motion, there is no acceleration in the horizontal direction. The acceleration, a, in the vertical direction is just due to gravity, also known as free fall

a_{x}=0

a_{y}=-g

### Velocity

The horizontal velocity remains constant, but the vertical velocity increases linearly, because the acceleration is constant. At any time, t, the velocity is:

Where v refers to velocity at time, t. You can also use the Pythagorean Theorem to find velocity:

### Displacement

At time, t, the displacement components are:

The equation for the magnitude of the displacement is

### Parabolic Trajectory

The equation of a parabola is

### Maximum Height

The maximum height is reached when vy=0 . Using this we can rearrange the velocity equation to find the time it will take for the object to reach maximum height

where t_{h} stands for the time it takes to reach maximum height.
From the displacement equation we can find the maximum height

### Range

The range of the motion can be found by using the following equation