Video Transcript
A car is initially at rest before
it starts to roll along a downward-sloping road with its engine turned off. While rolling, the car’s velocity
increased by 1.4 meters per second. What vertically downward distance
does the car travel? Gravity is the only force that acts
on the car.
Let’s say that this is our car at
the initial moment when it’s at rest before it starts to roll downhill. After rolling for some time, we’re
told that the car’s velocity is increased by 1.4 meters per second. We want to know what is the
vertically downward distance, we’ll call it 𝑑, that the car has traveled for this
change to take place. To solve for this distance, let’s
recognize that this scenario involves the conservation of mechanical energy.
In general, a system’s mechanical
energy equals the sum of its potential and kinetic energies. In our scenario, we’re working with
a closed system, one where energy is neither added to the system nor taken away. Along with this, the initial and
final energy of the car can be expressed purely in terms of mechanical energy. This means that the mechanical
energy of our system consisting of the car and the road is conserved. We can write then that the initial
mechanical energy of our system equals its final mechanical energy. We can expand this equation in
terms of initial and final potential and kinetic energies.
We’re going to say that the initial
moment in our system is when our car is positioned here at rest. The final moment we’ll say is when
it has achieved a speed of 1.4 meters per second downhill. Considering our car at its initial
position, we know that because it is at rest, it will have zero kinetic energy. Therefore, KE sub 𝑖 is zero. Likewise, if we choose to set the
elevation of the car at its final moment at a height of zero, then at this final
moment, the car’s gravitational potential energy will be zero. Since the car possesses no other
kind of potential energy, for example, spring potential energy, we can say that PE
sub 𝑓 is zero.
All of that then brings us to this
expression, the initial potential energy of our system, specifically gravitational
potential energy, equals its final kinetic energy. Clearing a bit of space on screen,
we can recall that in general, an object’s kinetic energy equals one-half its mass
multiplied by its speed squared. And along with this, an object’s
gravitational potential energy equals its mass times the acceleration due to gravity
multiplied by its height relative to some reference level. In our case, this height ℎ is the
distance 𝑑 we’re trying to solve for.
So then, based on this equation, we
can write that 𝑚 times 𝑔 times 𝑑, the car’s initial mechanical energy, equals
one-half 𝑚 times its final speed squared, its final mechanical energy. Note that in this equation, mass is
common to both sides and therefore can cancel out. If we then divide both sides by the
acceleration due to gravity 𝑔, that value cancels on the left and we find that 𝑑
equals 𝑣 squared over two times 𝑔. 𝑣 is equal to the final speed of
our car, 1.4 meters per second. And we remember that the
acceleration due to gravity is 9.8 meters per second squared. Therefore, 𝑑 equals 1.4 meters per
second all squared divided by two times 9.8 meters per second squared. This is exactly equal to 0.1
meters. That’s the vertically downward
distance this car needed to travel for its speed to increase by 1.4 meters per
second.
