"That's so Fetch!"

One of the best parts about going home for fall break was seeing my dog – she’s a Havanese, about a year old, and is always looking to play. Her favorite game is fetch, where I throw her toy and she runs, catches it, and brings it back. The physics is fetch is relatively simple – it involves motion in two dimensions: X (left and right) and Y (up and down).

I throw the toy at an angle Ø. Using trigonometry, we can calculate the initial velocity of the toy in the X direction (V_OX). In the diagram below, the toy is thrown to the right so V_OX = V_O * cos(Ø). The toy’s velocity in the X direction will remain the same throughout the entirety of the throw, as it no other forces interact with the toy in the X direction and Newton’s First Law tells us that every object in uniform motion (constant velocity), such as the toy, will continue in its uniform motion unless acted upon by another force.

The Y velocity is different. While you could calculate the initial Y velocity, once the toy is in the air, it’s acted upon by the force of gravity, which will affect the velocity. The acceleration due to gravity is 9.8m/s² towards the direction of Earth. For simplicity, we will say the acceleration in the Y direction is 9.8m/s² downwards for the entirety of the throw. Since the acceleration will affect the velocity in predictable ways, we can conclude some key information about the Y velocity of the toy. After being thrown, the toy will continue to rise upwards but slow down until it reaches its highest point, where V_Y = 0m/s. After that, the toy will fall and speed up until my dog catches it.

Putting the X and Y velocities together, we know that the toy follows a parabolic movement, as shown in the diagram below. If we had specific values for the initial velocity and angle of the throw, we could use the kinetic equations to calculate other variables such as the time the toy is spent in the air, the distance it travels, and even its final velocity. If I wanted to be an astute physics student, I could find the position of the ball and how long it will take to reach that position in order to catch the ball.

My dog’s never been taught the kinematic equations yet she manages to catch the toy. In her head, she can figure out where to go to catch the ball with only an initial velocity and angle to go off of. In theory, she then calculates the initial X velocity, time, and change in position. Even more, she only a second or two to do this and get in position to catch the toy. The change in position will be how far in the direction the toy will go in the amount of time it takes the toy to rise up then fall to my dog’s height, as she’s not flat on the ground. After all of this I’ve learned she’s a lot smarter than she seems.

One more caveat: Because extensive breeding for different traits, not all dog breeds are great at playing fetch. Here’s a cute video of a dog trying it’s best. Enjoy!