Quadratic Equations in Physics

In this post, you'll learn how initial velocities affect the trajectories of projectiles. This post contains interactive elements and is best experienced with a web browser at the original post.

Let's pull out our virtual slingshots for a moment and see just how far we can fling a pebble. A slingshot gives the pebble an initial velocity and the pebble is in freefall thereafter. Play with the velocity and see how the trajectory changes.

y' = `m / s`
x' = `m / s`
kinetic energy ~ Joules

What is happening here?

The shape of the trajectory stays the same no matter how hard it is thrown. By increasing x' the shape stretches out, and by increasing y' the shape shifts up. Still it looks more-or-less the same.

This shape is called a parabola, and it is generated by a quadratic equation. In fact, it is generated by this quadratic equation `t = 1/{x'} * x; y = g/2 * t^2 + y' * t,` where `t` is time and `g` is the constant acceleration due to gravity. Translated into ASCIIsvg JavaScript, it looks like this:

 g = -9.81;
 plot(function (x) {
     t = (1/b) * x;
     return g/2*t*t+a*t
 }, "cubic")
 

Where did that come from?

The quadratic solution follows directly from the assumption of constant acceleration.

`y'' = g`

From this we can integrate with respect to the time to find the vertical velocity. `y' = int g dt.` `y' = g t + C` where `C` is the initial velocity, since `y'(0) = C`.

Once we know the velocity across time, we can solve for the position. `y = int g t + y'(0) dt.` `y = g/2 t^2 + y'(0) t + C` where `C` here is the initial position, which we assume to be 0.

What about `x`? Gravity does not affect the `x` portion of the velocity, since it only accelerates downward. Therefore, `x'` is constant. Integrating, `x = int x' dt = x' t + C.` Since we can assume the slingshot shoots from the origin, `C` here is 0.

Note that `x` has been solved to be just a constant factor of the time `t.` Therefore, it makes sense to solve for the time first in the plots and then solve for the height.

g = -9.81;
plot(function (x) {
    t = (1/b) * x;
    return g/2*t*t+a*t
}, "cubic")

Where does the pebble hit the ground?

The pebble hits the ground when `y = 0.` In the previous sections we have already derived an equation of `y` with respect to time `t`. `y(t) = g/2 t^2 + y'(0) t.`

The first thing to do is to figure out when the pebble will hit the ground. Since `t` factors out, we needn't even pull out the quadratic equation. `y(0) = t (g/2 t + y'(0))` Thus. `t = 0` or `t = { -y'(0) 2/g } = { y'(0) 2/9.81 }` if `g = -9.81.`

Once we have when it hits, it is easy to find where it hits. Since the x-velocity remains constant, `x = x' t.` Just multiply the x-velocity by the time to find where!

This post was created as part of project 4 for Texas A&M Math 696. This project is based on the the ASCIIsvg example file.