Kinematics

At some point, someone thought that things would be a lot easier if he could stab someone else from a distance, and people have been optimizing projectile trajectories ever since. Pirates spruced up naval warfare with red-hot cannonballs [4]. King Edward I constructed a trebuchet so fearsome that Stirling Castle surrendered before any battle began, yet he knocked down a wall anyway (apparently medieval kings do not take “Yes” for an answer any more than they take “No”) [1]. These days, we’ve escalated to assaulting the cosmos itself. So far, we have successfully hit the Moon, Venus, Mars, a comet, and some asteroids [3].

While impressive, none of these projectile accomplishments match the surgical precision of my own sibling, who was capable of landing a Nerf bullet in my eye from any angle and through any obstacle. The plea that he was “shooting the air when [his sister] happened to get in the way” would have been more convincing had it not occurred with the reliability of a Swiss watch. He could have found his calling at NASA, shooting asteroids from the sky, except that one way or another, a space rock would have ended up in my face.

We have previously seen how to separate a vector (velocity) into its components by creating a right triangle. Trigonometry allows us to construct equations such as

where θ is the angle between the ground and the direction which the projectile is initially launched.

We also saw how Galileo rebutted Aristotle’s view of motion and proposed that free-falling objects obey the equations

where ∆x is the change in height, g is the gravitational acceleration of the Earth, t is time, and v_f is the final velocity the object reaches.

Galileo’s equations are excellent for an object that is falling directly towards the Earth and was dropped from rest (initial velocity is zero), but it fails to describe any of the motion which resulted in my eye being hit by a Nerf bullet. In my situation, the foam bullet was launched with an initial velocity that had both horizontal and vertical components. The bullet sailed through the air, changing speed and direction. Better tools to explain this motion are complete kinematic

equations:

where ∆x is change in position, v_i is initial velocity, v_f is final velocity, t is time, and a is acceleration [2]. These equations assume that the acceleration is constant, but the velocity can vary. By having multiple equations, each juggling more or less the same variables, it is possible to solve many problems, including projectile motion.

Had I the skill and practice of NASA computer Katherine Johnson, I could have taken the initial velocity and launch angle of my brother’s shot, determined the maximum distance it could have gone, the height at which the bullet would hit me, and how long it would take for me to meet my maker.

Let’s consider an example

If a rocket is launched at a 45-degree angle from the ground, with an initial velocity of 15 m/s , how far in the x-direction does the rocket go? How high in the y-direction does the rocket go at it’s peak?

References

[1] Susan Borowski. Trebuchets and their modern use | american association for the advancement of science (aaas). American Association for the Advancement of Science, May 2021. URL: https://www.aaas.org/taxonomy/term/10/trebuchets-and-their-modern-use.

[2] David Halliday, Robert Resnick, and Jearl Walker. Fundamentals of physics. Wiley, 10 edition, 2014.

[3] John M Logsdon. Solar system exploration. Encyclopedia Britannica, Sep 2024. URL: https://www.britannica.com/science/space-exploration/Solar-system-exploration.

[4] Melissa Petruzzello. Pirate school: 5 things you can shoot from a cannon. Encyclopedia Britannica. URL: https://www.britannica.com/story/5-things-you-can-shoot-from-a-cannon.