## Rolling Down a Hill

Given two cars are rolling down a hill. Car A weighs 10 kg, and Car B weighs 20 kg. Ignoring friction, which one would reach the bottom first? Why? Assume all environmental and design attributes are identical for both cars.

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__**Hint**

**Hint**

$$$force=mass\times acceleration$$$

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__**Hint 2**

**Hint 2**

Velocity is a derivative of acceleration.

In a vacuum, both cars would reach the bottom of the hill simultaneously. Remember:

$$$acceleration=\frac{force}{mass}$$$

and

$$$Force=mass\times gravity$$$

Combining the two equations, the masses cancel out:

$$$acceleration=\frac{mass(gravity)}{mass}=gravity$$$

We can verify this answer by plugging in some hypothetical values into the kinematic equation:

$$$\Delta x=v_0t+\frac{1}{2}at^2$$$

where
$$\Delta x$$
is the change in distance,
$$v_0$$
is the initial velocity,
$$t$$
is the time, and
$$a$$
is the acceleration.

Because all environmental and design attributes are identical, let's set the initial velocity to zero, and the distance to the hill's bottom to 2 meters:

$$$2=(0)t+\frac{1}{2}(gravity)t^2$$$

$$$2=\frac{1}{2}(9.8)t^2$$$

$$$\frac{4}{9.8}=t^2$$$

$$$t=\sqrt{0.41}=0.64\:sec$$$

Notice how the car's mass did not impact the calculation to determine how long it took each car to reach the hill's bottom.

Both cars would reach the bottom of the hill simultaneously due to friction being ignored.