Video Transcript
A wagon sits at the top of a
hill. The wagon is given a push that
increases its speed negligibly but is just sufficient to set the wagon in motion
down a straight slope. The wagon rolls 53.9 meters down
the slope, which is inclined 16.5 degrees below the horizontal, and reaches the
bottom of the hill. If friction is negligible, what
speed is the wagon moving when it reaches the bottom of the hill?
We can call the speed of the wagon
when it reaches the bottom of the hill ๐ฃ sub ๐. And weโll start on our solution by
drawing a diagram of the situation. A wagon sits perched at the top of
a long flat hill whose length weโve called ๐ฟ. Itโs 53.9 meters. The hill is inclined at an angle
below the horizontal โ weโve called ๐ โ given as 16.5 degrees. Weโre told the wagon is given a
very slight push to set it in motion and then rolls down the hill. And when it reaches the bottom of
the hill, we want to solve for its speed, ๐ฃ sub ๐.
In this scenario, our system
consists of the wagon and the hill. And since energy is neither added
to nor taken away from this system, we can say that itโs conserved. Applying the general statement of
energy conservation to this scenario, weโll use our freedom to decide when the
initial moment and when the final moments are. Choosing the initial moment to be
just when the cart is pushed at the top of the hill, and the final moment when it
just reaches the bottom of the hill.
We can expand this conservation
expression to say that the kinetic energy plus potential energy initially is equal
to the kinetic energy plus potential energy finally. At the outset, the wagon is not in
motion. So its initial kinetic energy is
zero. And if we choose the height at the
bottom of the hill to be our reference level of zero, we can say that the final
potential energy of the cart is also zero, since itโs at zero height. So the wagonโs initial potential
energy is equal to its final kinetic energy.
The wagonโs potential energy is
entirely gravitational. And we recall that gravitational
potential energy is equal to ๐ times ๐ times โ. Applying that relationship, we
write ๐ times ๐ times the initial height of the wagon equals KE sub ๐. We also recall that an objectโs
kinetic energy equals half its mass times its speed squared. We can write this in for our KE sub
๐ expression.
Looking at this equality, we see
that the mass of the wagon appears on both sides. So it cancels out. We want to solve for the final
speed of the wagon. So we rearrange this equation and
see the wagonโs final speed is equal to the square root of two times ๐ times โ sub
๐. The acceleration due to gravity,
๐, we take to be exactly 9.8 meters per second squared.
Looking at our diagram, we see that
the height of the hill, โ sub ๐, relates to the hill length, ๐ฟ, and the angle of
inclination, ๐. In particular, we can write that โ
sub ๐ is equal to ๐ฟ times the sin of ๐. So we substitute ๐ฟ sin ๐ in for โ
sub ๐ in our expression for ๐ฃ sub ๐.
Looking at this expression, weโre
given the length ๐ฟ, the angle ๐, and ๐ is a known constant. So weโre ready to plug in and solve
for ๐ฃ sub ๐. Entering these values on our
calculator, we find that, to three significant figures, ๐ฃ sub ๐ is 17.3 meters per
second. Thatโs the speed of the wagon when
it reaches the bottom of the hill.
