Video Transcript
A particle of mass 100 grams was projected vertically upward at 20 meters per second
from a point on the ground. Use the work–energy principle to calculate its kinetic energy when it was at a height
of 14 meters above the ground. Take 𝑔 equals 9.8 meters per second squared.
To start with, let’s recall the work–energy principle. The work–energy principle states that the total or net work done to some object is
exactly equal to the net change in its kinetic energy. Work itself is the energy that an object gains or loses as it moves in the same
direction or the opposite direction to some applied force. In particular, for constant forces parallel to the direction of motion, the work done
is the magnitude of the force times the distance over which the object moves. If we have several such forces, then the net work is just the sum of the
force-times-distance contributions from each individual force.
The total change in the kinetic energy of an object is the difference between its
final kinetic energy and initial kinetic energy. Furthermore, at any particular moment, the kinetic energy of an object can be
expressed as one-half times its mass times the square of its speed. If the object’s mass is constant, then its final kinetic energy is one-half times its
mass times the square of its final speed and the initial kinetic energy is one-half
times its mass times the square of its initial speed.
Okay, now, let’s figure out how to use all these equations to find what we’re looking
for. Our end goal is to find the kinetic energy of our particle at a height of 14 meters
above the ground. We can treat this as the final kinetic energy in the work–energy principle. We’re also told the mass of the particle as well as its initial speed, which is what
we need to calculate the kinetic energy of the particle, specifically its initial
kinetic energy. So, we have all the information we need to evaluate the right-hand side of the
work–energy principle.
To evaluate the left-hand side, the net work, we need to know the forces acting on
the particle as well as the distance it traveled. Well, we know the distance right from the statement. We’re told that the particle started on the ground, and we are interested in the
particle when it is 14 meters above the ground. So, the distance the particle traveled is 14 meters. The only force acting on the particle over this distance is the force of gravity.
The force of gravity, which we can express as 𝐹 sub 𝑔, is equal to the mass of an
object times the gravitational acceleration. We’re given the mass as we saw before, and we’re given a value for the gravitational
acceleration, 9.8 meters per second squared. Furthermore, because the particle was projected vertically upward and gravity always
pulls vertically downward, the force and direction of motion are parallel, which
means we can use force times distance to calculate the work.
Okay, let’s get to calculating. We have that the work done by gravity is equal to the final kinetic energy, which is
what we’re looking for, minus the initial kinetic energy. For the work due to gravity, we have the negative of the mass of the particle times
the gravitational acceleration times the distance traveled. This quantity is negative because the direction of the force of gravity is opposite
to the direction of the motion, which is vertically upward. If the object were instead falling downward, then the direction of gravity and the
direction of motion would be the same and the work would be positive. This gives us negative 100 grams times 9.8 meters per second squared times 14
meters.
Before plugging these numbers into a calculator, note that our units for distance are
meters and our units for time are seconds. However, our mass is given in units of grams not kilograms. Since our final answer will be in terms of joules, we need to use kilograms, meters,
and seconds, not grams, meters, and seconds. So, we need to convert grams to kilograms. Since one kilogram is exactly 1000 grams, 100 grams is one-tenth of one kilogram. So, the work is negative 0.1 kilograms times 9.8 meters per second squared times 14
meters. Plugging this into a calculator gives us negative 13.72 with the unit of energy
joules.
The initial kinetic energy is one-half mass times the square of the speed, which,
using 0.1 kilograms for the mass and 20 meters per second for the initial speed,
gives us one-half times 0.1 kilograms times 20 meters per second squared or 20
joules. Now that we know the work done by gravity and the initial kinetic energy, we’re ready
to solve for the final kinetic energy.
To solve for the final kinetic energy, we’ll add the initial kinetic energy to both
sides. KE 𝑖 minus KE 𝑖 is zero on the right-hand side. So, we’re just left with the final kinetic energy. On the left-hand side, we have the work done by gravity plus the initial kinetic
energy. Now, we plug in our numbers. Negative 13.72 plus 20 is exactly 6.28. Including our unit of energy, we find that 6.28 joules is exactly the kinetic energy
that we’re looking for.
