# Physics with Calculus/Mechanics/Projectile Motion

## Projectile Motion[edit]

Using the equations we derived in the last section, we can now use them to model the motion of a *projectile*. A projectile is an object upon which the only force acting is gravity, which means that in all situations, the acceleration in the *y* direction, . For simplicity, we will assume that the path of a projectile, also called its *trajectory*, will always be in the shape of a parabola, and that the effect of air resistance upon the projectile is negligible.

**The Horizontal Motion**

Since the only force acting upon the object is gravity, in the *y* direction, there is no acceleration in the *x* direction.

Let us assume that the projectile leaves the origin at time *t* = 0 and with speed *v _{i}*. Then we have a vector

**v**that makes an angle of θ

_{i}_{i}with the

*x*-axis. Then, using a bit of trigonometry, we have the following:

and

Rearranging for the initial velocities, we get the initial *x* and *y* components of velocity to be

- and

**The Vertical Motion**

There is a constant acceleration down, g which is the force of gravity. Accelecration is the instantaneous rate of change of velocity so:

therefore we can integrate acceleration with respect to time to get velocity

Velocity is the instantaneous rate of change of displacement so:

We can also integrate velocity with respect to time to get displacement