Here is a question. It’s a sort of subtle question, but one that can be answered with freshman-level physics. But it’s an excellent test of understanding. I’m not promising that the question itself is not in some sense a “trick” question, but the trick is in how you might think about the physics, not the question itself. Which is to say, I’m not promising that what the question assumes happens can actually happen. But it might – it’s up to you to tell me!
Ok. You have a perfectly efficient car, which transmits all of the energy in its fuel into kinetic energy of forward motion. Kinetic energy is related to the forward velocity by the famous equation
We’ll make up our own unit of energy (the zap) so the concept is more clear and we don’t have to worry about the particular mass of the car. The driver of the car pours 1 zap worth of fuel into the engine and the car accelerates from rest to 1 m/s. To double the speed you have to quadruple the kinetic energy. So you pour three more zaps of fuel into the car, and with a total kinetic energy of 4 zaps the car is moving at 2 m/s. To go 100 m/s, the car needs 10,000 zaps of fuel.
But what if you’re watching the experiment from a bus which is itself traveling at 100 m/s alongside the car, then you observe the car standing stationary. One more zap of energy should by your prediction therefore increase its speed in your reference frame by 1 m/s – or equivalently 101 m/s with respect to the ground. But from the reference frame of the ground, the speed of the car should be sqrt(K), where K is the 10,001 zaps of energy the car has used in total. That should put the car at 100.005 m/s, not 101 m/s.
Who is right, given that we know both inertial reference frames are equivalent?