Starting from rest, several toy cars roll down ramps of differing lengths and angles.

Question

Starting from rest, several toy cars roll down ramps of differing lengths and angles. Rank them according to their speed at the bottom of the ramp, from slowest to fastest. Car A goes down a 10 m ramp inclined at 15 ∘ , car B goes down a 10 m ramp inclined at 20 ∘ , C goes down an 8.0 m ramp inclined at 20 ∘ , and car D goes down a 12 m ramp inclined at 12 ∘ .

Please enter the letters of the cars in order from slowest to fastest, separated with commas.

Answer 1

Let

m be the mass of a car,

L be the length of its ramp,

α be the angle of the ramp's incline,

g be gravitational acceleration,

v be the car's speed at the bottom.

The speed v at the bottom determines the car's kinetic energy.

That kinetic energy equals its initial potential energy.

That potential energy depends on the starting height.

So the height is the indicator of the final speed v.

Let's look in detail at the energy transfer from potential to kinetic.

Lsin(α) is the height from which the car starts.

mgLsin(α) is the car's initial potential energy.

mv²/2 is the car's final kinetic energy.

By conservation of energy

mv²/2 = mgLsin(α)

From that express

v = √(2gLsin(α))

Notice that g is a constant, common to all the cars.

The final velocity is independent of the car's mass m.

Therefore the only way a car can increase its final velocity is by increasing the product

Lsin(α)

So, simply calculate this product for each car, and their values will determine the ranking.