If two vehicles are identical, except for one weighing twice as much as the other vehicle, and they roll down a hill, which car reaches the bottom first?

Neglect air resistance. Assume the extra mass is in the body of the vehicle

Hint: What governing physics law is used to characterize this system?

EDIT: Please refer to @bmachadoโ€˜s post. The answer below provides context but makes incorrect assumptions regarding increased kinetic energy proportional to increased velocity.

The governing equation for a vehicle rolling down a hill regardless of the weight is:

mgh = 1/2mv^2 + 1/2Iw^2

The first term is the potential energy; this is the energy is takes to lift the object up the ramp. This is equal to ๐‘š๐‘”โ„Ž with ๐‘šm being the mass, ๐‘” the acceleration due to gravity, and โ„Ž the height of the ramp.

The second term is the translational kinetic energy; this is the energy it takes for the object to move down the ramp.

The third term is the rotational kinetic energy; this is the energy it takes for the object to roll. This is equal to 1/2๐ผ๐œ”^2, with ๐ผ being the moment of inertia (the objectโ€™s resistance to being rotated) and ๐œ” being the angular velocity.

Itโ€™s important to note that the m term:

When you increase the weight, the variable being affected is m . The mass variable affects either the translational kinetic energy (m term) OR the rotational kinetic energy (I term, moment of inertia is proportional to m).

Moment of Inertia usually follows the form like this:

I = mr^2

but with a constant scaling factor. m here would be the mass of the wheel, not the entire vehicle itself.

It all depends on the location of the additional mass. Discussion has raised good points so hopefully the answer can be hashed out. Overall though, employers are just looking for you to explain your thought process and have a good foundation of the underlying principles. Theyโ€™re not testing for the exact answer, but your thought process.

Here are some useful links!

The basic idea is since weight location is not specified, extra mass could either go into increasing vehicle total mass (m) or moment of inertia (I)

I dont think this is correct. If the wheel geometry and mass are equal for both cars, then that term will be the same quantity for both. This means the translational term will increase for the car with the higher mass, but that term is the total energy caused by translational energy, not just the speed. The larger mass of the body will add more energy at the bottom of the hill, but it will not increase the speed.

Not 100% sure about what I said above, but it is what I am logically thinking. Thoughts?

EDIT: I did an example problem below

Comments about this solution:

  • I did not round anything, and the V^2 term was slightly different, this means it actually did have a slightly faster speed, but very very small?
  • My assumption about the wheels taking the same energy might not be true? how can we know if it is true?
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EDIT: Math errors, mass of vehicle should cancel out for both potential energy and kinetic energy terms. Great catch @bmachado, I agree with your solution.

This is a tough question and I was not entirely sure thinking through the answer. Iโ€™ll mark your reply as the solution and make edits to the original reply.

Watch the video at 05:50. This might help.

Thatโ€™s a great visual representation to understand! Thank you @gowtham for the help. :slight_smile: