Physics with Calculus/Mechanics/Projectile Motion

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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 vi. Then we have a vector vi that makes an angle of θi with the x-axis. Then, using a bit of trigonometry, we have the following:


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


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