A simple physics question was asked yesterday at work, but it generated a lot of debate: “A guy’s on top of a 20 foot ladder when it falls over. How fast was he going when he hit the ground?”
For simplicity, ignore issues such as the foot of the ladder moving, air resistance, seats, humidity, etc. Also assume the guy doesn’t fall off the ladder, but follows it down. (This is as realistic as a spherical, frictionless, non-conducting bovine.)
If our friend was just dropped from 20′ above the ground, we could calculate the time to fall as:
Distance = Average velocity * time
Which works out to:
20 feet = 1/2 * 32 feet/s2 * t2
or, solving for time:
=Time = Distance / Average velocity
which is:
Time = sqrt [ 2 * (20/32) ]
T = 1.12 seconds
Velocity at impact would be:
Velocity = Acceleration * time
= 32 ft/s2 * 1.12 s
= 35.8 ft/s, or 24.3 miles per hour
All this is straightforward. The controversy came in because our friend is traversing an arc of 31.4 feet through space. What changes?
Answer: nothing.
Using the calculations from Amusement Park Physics, this is just a pendulum. We know the potential energy at the top, and the kinetic energy at the bottom. With the same assumptions as above, we can solve for velocity.
Kinetic Energy, Ek = 1/2 * mass * velocity2
Potential Energy, Ep = mass * G * heightEk = Ep
1/2 * m * v 2 = m * G * h1/2 *
m* v 2
=m* G * h
= sqrt ( 2 * G * h)
= sqrt ( 2 * 32 feet / s2 * 20 feet)
= 35.8 ft/s, or 24.3 miles per hour
Practically speaking, our friend broke his leg.
There were numerous creative ways to solve the problem the wrong way. And as an engineer who hasn’t taken physics in nearly 20 years, I pursued a couple of these before realizing this is the correct answer.
A more interesting extension is the reality where the guy falls off the ladder and becomes a projectile. This is too much to think through right now.