Applying the Work-Energy Theorem to Falling Objects

So far we’ve applied the Work-Energy Theorem to a flying object, namely, Santa’s sleigh, and a rolling object, namely, a car braking to avoid hitting a deer. Here we’ll apply the Theorem to a falling object, that coffee mug we’ve been following through this blog series. We’ll use the Theorem to find the force generated on the mug when it falls into a pan of kitty litter. This falling object scenario is one I frequently encounter in previous topics and it’s something I’ve got to consider when designing objects that must withstand impact forces if they are dropped.

Here’s the Work-Energy Theorem formula again,

F × d = ½ × m × [v₂² – v₁²]

where F is the force applied to a moving object of mass m to get it to change from a velocity of v₁ to v₂ over a distance, d.

As we follow our falling mug from its shelf, its mass, m, eventually comes into contact with an opposing force, F, which will alter its velocity when it hits the floor, or in this case a strategically placed pan of kitty litter. Upon hitting the litter, the force of the mug’s falling velocity, or speed, causes the mug to burrow into the litter to a depth of d. The mug’s speed the instant before it hits the ground is v_1,and its final velocity when it comes to a full stop inside the litter is v₂, or zero.

Inserting these values into the Theorem, we get,

F × d = ½ × m × [0 – v₁²]

F × d = – ½ × m × v₁²

The right side of the equation represents the kinetic energy that the mug acquired while in freefall. This energy will be transformed into Gaspard Gustave de Coriolis’ definition of work, which produces a depression in the litter due to the force of the plummeting mug. Work is represented on the left side of the equal sign.

Now a problem arises with using the equation if we’re unable to measure the mug’s initial velocity, v₁. But there’s a way around that, which we’ll elaborate next time when we put the Law of Conservation of Energy to work for us to do just that.