Why does a ball roll faster than a model car down a ramp?

[Heyxyz]

Homework Statement

A toy car and a solid metal sphere are rolled down an incline, why does the car reach the bottom first?

Relevant Equations

KE (total) = KE + KE (rotational) = 1/2mv^2 + 1/2Iw^2

Sphere: I = 2/5mr^2

v = wr

w = omega

Hello,

I tried to put it in an equation, but it didn't really work out. In this situation, the car was about the size of a model, and, while not exact, the radius of each wheel couldn't have been more than like a centimeter. Conversely, the ball was like twice the size of the car and had a diameter of 10 - 15 cm.

Ball:

=1/2mv^2 + 1/2I(V/R)^2, I = 2/5mr^2

= 1/2mv^2 + (1/2)(2/5)(m)(r^2)(v^2/r^2)

=7/10mv^2

Car:

Since wheels are practically cylinders, I figured I = 1/2mr^2. There are four wheels, so 4 * 1/2mr^2 = 2mr^2

=1/2mv^2 + 1/2I(V/R)^2, I = 2mr^2

= 1/2mv^2 + 2m(r^2)(v^2/r^2)

=1/mv^2 + mv^2

= 3/2mv^2I don't understand where I'm going. If I follow my above equations, it would appear that the ball reaches the bottom sooner due to the smaller coefficient, but I know that isn't true. Maybe the smaller radius of each wheel has something to do with it?

Thank you.

 

[haruspex]

Since wheels are practically cylinders, I figured I = 1/2mr^2

I doubt the car body is massless. I would take it as being far more massive than the wheels.

 

[Heyxyz]

That's true. Does the car's mass make a difference? I know mass is there, but I just figure it could be ignored (we weren't given masses). I might sound like a fool, but I don't even know how an external mass such as the car would affect the wheels of their velocity.

 

[haruspex]

That's true. Does the car's mass make a difference? I know mass is there, but I just figure it could be ignored (we weren't given masses). I might sound like a fool, but I don't even know how an external mass such as the car would affect the wheels of their velocity.

The weight of the car is the primary driving force, much more important than the weight of the wheels. And the mass of the car does not rotate, so all its KE goes into linear motion.
Consider the wheels to be so light in relation to the car that they are irrelevant.

 

[Heyxyz]

Okay. In other words, the wheels essentially exist so we can ignore friction, and my car's energy should be based on the body of the car rather than the wheels. That actually makes a lot of sense. Thank you.

 

[haruspex]

Okay. In other words, the wheels essentially exist so we can ignore friction, and my car's energy should be based on the body of the car rather than the wheels. That actually makes a lot of sense. Thank you.

Correct.