Scientific Programming II

ModSim

Unit 2 - Physics Explanation

Two-Dimensional Motion

So far we have discussed particles moving in a single dimension (represented by its velocity vector). However, objects can move in multiple dimensions in the real world. Galileo was one of the first scientists to postulate that two-dimensional motion is actually two, one-dimensional motions superimposed on each other.

Therefore, we can say for any object traveling ballistically (e.g., without thrust or power):

• v⃗ = v⃗_x + v⃗_y

Now the question is how to find both v⃗_x and v⃗_y.

Let's imagine a football being kicked towards the goalposts. Since the football has no engine or rockets on it, it is in a free, or ballistic, flight. The kicker will kick the football at some initial angle θ, and since we know that the motion is actually two separate motions in the x- and y-directions, we can find the initial velocities in both the x- and y-directions. Accepting that the calculus is accurate, it can be said that:

• v⃗_iy = v_isin(θ)

• v⃗_ix = v_icos(θ)

• s_x = v_ixt

• s_y = v_iyt + 0.5gt²

Now, we need to find some more properties of the flight. Two of the most important are the total flight time t^T and peak height s^p. The total flight time is the amount of time for a projectile to reach the same height from which it was launched. The peak height is the greatest height achieved by the projectile. The equations can be derived (see a textbook), but are given below:

• t_T = -(2 * v_iy) / g

• s_p = -(v_iy)² / 2g

The range of a projectile is the distance it travels before returning to the same height from which it was launched. It can be found by substituting the equation for total time into the equation for displacement in the x-direction, giving us:

s_R = -(2v_i² / g) * cos(θ)sin(θ)