Physics with Calculus/Mechanics/Projectile Motion

Projectile Motion

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, $a_y=-g$. 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_i that makes an angle of θ_i with the x-axis. Then, using a bit of trigonometry, we have the following:

$\cos {\theta }_i=\frac{v_{xi}}{v_i}$

and

$\sin {\theta }_i=\frac{v_{yi}}{v_i}$

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

$v_{xi}=v_i\cos {\theta }_i$ and $v_{yi}=v_i\sin {\theta }_i$

$x=v\cos (\theta )t$

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:

$\frac{dv}{dt}=a=g$

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

$\int dv=\int gdt$

$v=gt$

Velocity is the instantaneous rate of change of displacement so:

$\frac{dd}{dt}=v$

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

$\int dx=\int vdt$

$\int dx=\int gtdt$

$d=\frac{1}{2}g{t}^2$