Playing: Killing Inanimate Objects with Kinematics

I was inspired to look more closely at cat physics while playing with my cat, Calvin, during a study break. Physics drives everything that he does, and an energetic cat is especially well-suited to kinematics problems. For all of these problems, the cat's mass is 10.5 lbs = 4.8 kg. We are ignoring air resistance, which may not be the best idea since he is fluffy.

Linear Motion...

Calvin starts at rest and spontaneously decides to zoom across the apartment. While cats have a maximum speed of 48 km/h, obstacles indoors restrict his speed to a more modest 12 km/h. He reaches this top speed by the time he leaves the room, having sprinted 3 m. What is his acceleration and the magnitude of the force causing that acceleration?

Assuming constant acceleration: acceleration = (final_velocity^2 - initial_velocity^2) / (2 * distance).

Calvin's final velocity is given as 12 km/h * 1000 m / 1 km * 1 hour / 3600 seconds = 3.3 m/s. He started at rest so his initial velocity = 0 m/s and it is stated that he sprinted distance = 3 m.

Plugging those values in we get acceleration = [(3.3 m/s)^2 - (0 m/s)^2] / (2 * 3 m) = (10.89 m^2/s^2) / 6 m = 1.815 m/s^2

Which, combined with Calvin's stated mass, lets us solve force = mass * acceleration = 4.8 kg * 1.815 m/s^2 = 8.712 N.

...And Collision

While zooming across the apartment, Calvin sees a 7 g feather toy being dragged across the ground 1 km/h perpendicular to his path. He catches the feather toy and yanks it and the 60 g wand it is attached to out of his owner's hand. What is his speed and direction as he drags his kill away?

Calvin and his "prey" are moving perpendicular to each or at an angle of 90 degrees, so we can say that Calvin is moving along the x-axis and the feather toy is moving along the y-axis. Since he catches the toy and drags it away, their collision is inelastic. The formula for their combined momentum is (Calvin's_mass + toy's_mass) * final_velocity = (Calvin's_mass * Calvin's_initial_velocity) + (toy's_mass * toy's_initial_velocity). We'll solve for the x- and y-final velocities separately.

x-final_velocity = final_velocity * cosine_theta = (Calvin's_mass * Calvin's_x-initial_velocity) + (toy's_mass * toy's_x-initial_velocity) / (Calvin's_mass + toy's_mass). Calvin was zooming along the x-axis at 3.3 m/s but the toy was moving along the y-axis so its x-initial_velocity is 0 m/s. Calvin's mass was given earlier as 4.8 kg and the 7 g toy plus its 60 g wand weigh 0.067 kg combined.

We plug those values into the x-final_velocity formula and we get x-final_velocity = final_velocity * cosine_theta = [(4.8 kg * 3.3 m/s) + (0.067 kg * 0 m/s)] / (4.8 kg + 0.067 kg) = 15.84 kg m/s / 4.867 kg = 3.25 m/s.

y-final_velocity = final_velocity * sine_theta = (Calvin's_mass * Calvin's_Y-initial_velocity) + (toy's_mass * toy's_Y-initial_velocity) / (Calvin's_mass + toy's_mass). This time, Calvin's initial velocity is 0 m/s while the toy was being dragged along the y-axis at 1 km/h * 1000 m / 1 km * 1 hour / 3600 seconds = 0.278 m/s.

y-final_velocity = final_velocity * sine_theta = [(4.8 kg * 0 m/s) + (0.067 kg * 0.278 m/s)] / (4.8 kg + 0.067 kg) = 0.018626 kg m/s / 4.867 kg = 0.00383 m/s.

Now that we have final_velocity * cosine_theta = 3.25 m/s and final_velocity * sine_theta = 0.00383 m/s, we can solve for theta = arctan[(final_velocity * sine_theta)/(final_velocity * cosine_theta)] = arctan(0.00383 m/s / 3.25 m/s) = arctan(0.001178462) = 0.06752 degrees.

And final_velocity = x-final_velocity / cosine_theta = 3.25 m/s / cosine(0.06752 degrees) = 3.250002257 m/s.

In other words, Calvin snatches up the toy and continues forward moving only 0.05 m/s slower.