Worksheet
Advanced Properties of Freely Falling Bodies #1
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As we continue our study of freely-falling bodies, there are further properties that we should understand to more fully describe a projectile's behavior.
In addition to its kinematics properties of how fast (velocities), how far (heights and displacements), how long (time) there are the properties of momentum, potential energy, and kinetic energy. Moreover, we are often asked how much work did gravity do on an object during the object's trajectory.
Momentum
is a vector quantity calculated as the product of the object's mass times its velocity. The direction of the momentum vector agrees with the direction of the object's velocity.
The formula to calculate momentum is:
p
= m
v
.
Momentum is measured with the units of kg m/sec.
Potential energy
is a measure of the stored energy an object acquires by virtue of its position in a gravitational field. Potential energy is calculated as the product of an object's mass times the local gravitational field strength times the object's height above an arbitrarily chosen zero position. Potential energy is a scalar quantity.
The formula to calculate potential energy is: PE
_{g}
= mgh.
Potential energy has units of kg(m/sec
^{2}
)(m) = kg m
^{2}
/sec
^{2}
= Joules
Kinetic energy
is a measure of the energy acquired by an object due to its motion. Kinetic energy is also a scalar - that is, it only has magnitude, not direction. Kinetic energy is calculated as ½ of the product of the object's mass times the square of the object's velocity.
The formula to calculate kinetic energy is: KE = ½mv
^{2 }
.
Kinetic energy has units of kg (m/sec)
^{2}
= kg m
^{2}
/sec
^{2}
= Joules
Work
is yet another scalar quantity. The work done on an object is defined as the product of the applied force which is parallel (or anti-parallel) to the direction of motion times the distance through which the object is moved. The work done on an object is also defined as the change in an object's kinetic energy. There are two formulas to calculate work.
These formulas are: W = F
_{||}
d and W = KE
_{f}
- KE
_{o}
.
Work is also measured in Joules.
When the work done on an object is positive, the object has gained speed and therefore KE. Negative work means that the object has slowed down and lost KE.
A
freebody diagram (FBD)
is a diagram summarizing only the forces acting on the object. In this unit on freely-falling bodies, we will examine two forces: weight and normals. We have already covered
weight
(the force of gravitational attraction between the mass and the mass of the planet). A
normal force
is a force supplied by a supporting surface. This force can be supplied by someone's hand holding a projectile, a table and the floor/ground.
The weight vector is always drawn by starting at the object's center of mass and pointing straight to the center of the earth/planet. The normal is always drawn perpendicular to the plane of the supporting surface and also passes through the center of mass. When an object is in
equilibrium
these two vectors will have exactly the same magnitude, or length. This would occur if the object were at rest or moving at a constant velocity (no acceleration). If an
unbalanced force
acts on the object (for example, only one force) then the object will be accelerated in the direction of that unbalaced force. An example of this situation would be a projectile in freefall.
Refer to the following information for the next six questions.
A 24.5 N ball is held at rest 10 meters above the surface of the earth.
Find the values of the following givens stated in the problem:
1(a). Height (h) = _______ m
1(b). Weight (Wt) = _______ N
2. Draw a freebody diagram (FBD) of the forces acting on the ball while it is being held at rest in your hand.
3. What is the gravitational field strength of the earth at the ball’s location given that the mass of the earth is 6.0 x 10
^{24}
kg and the earth’s average radius is 6.4 x 10
^{6}
meters?
4. What is the mass of the ball?
5. How much potential energy does the ball have, relative to the ground, while it is being held 10 meters above the earth's surface?
Refer to the following information for the next ten questions.
To release the ball your hand is gently removed thus allowing the ball to fall to the earth from a state of rest.
Find the values of the following givens stated in the problem.
6(a). Initial velocity (v
_{o}
) = _______ m/sec
6(b) Displacement (s) = _______ m
7. Draw a FBD of the ball while it is falling if the force of gravitational attraction of the earth is the only force acting on the ball.
8. Based on your answer to
Question 3
, what acceleration will the ball experience as it falls to the earth’s surface?
9. What is the ball’s velocity just as it strikes the ground?
10. How much time did it take the ball to reach the ground?
11. If a 4.9 N brick had been dropped from rest instead of the ball, would the brick require more, less, or the same amount of time to reach the ground? Support your choice.
more time
less time
the same amount of time
12. How much momentum will the ball have just as it strikes the ground?
13. How much kinetic energy will the ball have just as it strikes the ground?
14. How much work did gravity do
on the ball
as the ball was pulled to the earth’s surface?
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