Video Transcript
A child with a mass of 36 kilograms
carries a sled with a mass of 14 kilograms to the top of an evenly sloping hill,
walking 33 meters along the hillside and moving vertically upward by 8.8 meters. The child puts the sled on the
slope, where it is just held in place by friction, and carefully climbs on
board. The added weight of the child is
just enough to start the sled moving, and it slides down the hill, moving at 10
meters per second when it arrives at the base of the slope. How much energy is dissipated
during the sled’s downhill motion?
To work out how much energy is
dissipated when the sled goes down the hill, we need to compare the mechanical
energy of the child and the sled before and after this motion. Recall that the mechanical energy
of an object is equal to the sum of its gravitational potential energy and its
kinetic energy. As the child sleds down the hill,
some of the mechanical energy of the child and sled is dissipated by friction. This is because friction is a
force, which does work to decrease the kinetic energy of an object. The amount of energy that is
dissipated by friction is equal to the initial mechanical energy of the child and
the sled at the top of the hill minus the final mechanical energy of the child and
sled at the bottom of the hill.
Let’s start by looking at the
initial mechanical energy of the child and the sled, which is before they go down
the hill. We can calculate the gravitational
potential energy, hereafter GPE, of the child and sled, using the formula GPE equals
𝑚𝑔ℎ, where 𝑚 is their combined mass. 𝑔 is the gravitational field
strength. And ℎ is their height above the
ground. The value of 𝑚 is equal to the
mass of the child, 36 kilograms, plus the mass of the sled, 14 kilograms. This gives us a total mass of 50
kilograms. We have to add these two masses
because the child and the sled always move together as if they were one object. The gravitational field strength,
𝑔, is equal to 9.8 meters per second squared.
Now, when looking at the height, ℎ,
we’re given two distances. We’re told that the child walks 33
meters along the hillside, moving a total distance of 8.8 meters vertically
upwards. When calculating the gravitational
potential energy of the child and the sled, we’re only interested in the vertical
height above the ground, which is 8.8 meters. The distance of 33 meters is
irrelevant to the gravitational potential energy. So, the gravitational potential
energy of the child and the sled is equal to 50 kilograms multiplied by 9.8 meters
per second squared multiplied by 8.8 meters. This gives us a value of 4312
joules.
The child and the sled are both
stationary at the top of the hill, so the kinetic energy is zero joules. So, the mechanical energy of the
child and the sled at the top of the hill, the initial mechanical energy, is equal
to 4312 joules plus zero joules, which just equals 4312 joules.
Now, let’s calculate the mechanical
energy of the child and the sled at the bottom of the hill, the final mechanical
energy. When the child and sled arrive at
the base of the slope, their height above the ground is zero. This means that their gravitational
potential energy is zero. But when they reach the ground,
they have a speed of 10 meters per second. This means that they now have some
kinetic energy, hereafter KE, which we can calculate using the formula KE equals
one-half 𝑚𝑣 squared, where 𝑚 is the combined mass of the child and the sled,
which we know is 50 kilograms, and 𝑣 is their speed, 10 meters per second.
If we substitute these values into
the formula, we find that the kinetic energy is equal to one-half times 50 kilograms
times 10 meters per second squared. This gives us a value of 2500
joules for the kinetic energy. So, the mechanical energy of the
child and the sled at the bottom of the hill is equal to zero joules plus 2500
joules, which is just 2500 joules.
So, before the sled moves down the
hill, the mechanical energy of the child and the sled is 4312 joules. After the sled has moved down the
hill, the mechanical energy is 2500 joules. The energy dissipated during this
motion is equal to the initial mechanical energy, 4312 joules, minus the final
mechanical energy, 2500 joules. Completing this calculation, we
find that 1812 joules of energy are dissipated during the sled’s downhill
motion. So, the final answer to this
question is 1812 joules.
