The following Activities are for the 'Work and Energy in Free Fall' applet. Make sure you know how the applet functions by consulting Help and ShowMe under Applet Help on the applet's Help menu.
Activity 1. The purpose of Activity 1 is to use the work-kinetic energy theorem to predict the height a ball will reach that is thrown upward with a given speed at a given angle. Air resistance is ignored.
RESET the applet. Set the following parameters and the following initial conditions at t = 0 for the ball:
Outline
(a) Calculate KE(0).
(b) Calculate KEtop and DKE.
(c) Compare the work W shown in the Data box to DKE.
(d) In your Notebook, draw both the ball's displacement and
the force of gravity acting on the ball as the ball goes to the
peak of its trajectory. On the basis of this diagram, determine
the sign of W.
(e) From the equation, W = DKE, calculate the height h of
the ball's trajectory.
(f) Repeat all of the above for a different mass.
Details
(a) Calculate the ball's kinetic energy KE(0) at t = 0. Compare your result with the value shown in the Data box.
(b) Calculate the ball's kinetic energy KEtop at the peak of the ball's trajectory. Hint: The horizontal component of the ball's velocity is constant and equal to the ball's speed at the peak.
Determine KEtop 'experimentally' with the applet. To get a good value, STEP towards the peak of the trajectory using a sufficiently small time step setting. Watch the ball's elevation y change in the Data box. If you overshoot, REWIND the applet and try again.
Does your experimental value of KEtop agree with the value you have calculated?
What is the change DKE in the ball's kinetic energy from the start of the motion to the peak of the trajectory?
(c) The Data box shows the gravitational work W done since the beginning of the motion. W is equal to the net work done on the ball because the force of gravity is the only force acting on the ball.
The work-kinetic energy theorem requires that net work done on the ball up to a given instant is equal to the ball's kinetic energy change DKE up to that instant. Can you confirm that this is true at the peak of the ball's trajectory when you compare your value of DKE to the value of W shown in the Data box?
(d) At the peak of the trajectory, or as near to it as you can get, display the ball's displacement vector from the start to the peak and the force of gravity acting on the ball. Draw both vectors in your Notebook, tail-to-tail. Observe that the vectors form an angle greater than 90o between them, which means that the work W done on the ball by the gravitational force during this displacement is negative. This, in turn, means that the ball slows down as it goes up.
(e) Since the x-component of the force of gravity is equal to zero,
W = Fxdx + Fydy = 0 + (-mg)dy (1)
where
dy = h - 10 m (2)
is the y-component of the ball's displacement and y = h the ball's elevation when the ball is at the peak of its trajectory.
By the work-kinetic energy theorem, W should be equal to the change DKE in the ball's kinetic energy as the ball is going from y = 10 m to y = h.
Thus,
W = -mg(h - 10 m) = DKE. (3)
Note that in the present case both sides of Eq.(3) are negative.
Use Eq.(3) and the value of DKE found earlier to calculate h. Then use the applet to determine h 'experimentally'. Compare the experimental value of h to the calculated value.
(f) Repeat all of the above for a different mass, m = 400 g. Do you obtain a different value for h?
Explain your result theoretically by replacing DKE in Eq.(3) by (m/2)D(v2). Notice that the factor of m occurs on both sides of the equation and therefore can be canceled. Projectile motion without air resistance does not depend on the mass of the object.
Activity 2. The purpose of Activity 2 is to use the work-kinetic energy theorem to predict the speed of a ball when the ball reaches the ground after having been thrown from a given point above the ground with a given speed at a given angle. Air resistance is ignored.
RESET the applet. Set the parameters and the initial conditions at t = 0 for the ball to the same values as in Activity 1:
Outline
(a) Calculate KE(0).
(b) Play the motion until the ball reaches the ground at
y = 0, observe both the displacement of the ball and the
force acting on the ball, and determine the sign of the work
W.
(c) Calculate W, and compare it to the value shown in
the Data box.
(d) Equate W to DKE,
and determine KEground.
(e) From KEground, determine
vground.
(f) Repeat all of the above for a different mass.
Details
(a) Use the value KE(0) of the ball's kinetic energy at t = 0 from Activity 1. If you did not do Activity 1, calculate this value and compare your result with the value shown in the Data box.
(b) PLAY the motion the motion, and PAUSE it when the ball reaches the ground level, which we will take to be at y = 0. Display both the ball's displacement and the force of gravity acting on the ball. Observe that both vectors point in the same direction, downward, which means that the work done on the ball by the force of gravity while the ball is moving from y = 10 m to y = 0 is positive.
Therefore, since the work done by gravity is the net work done on the ball, the work-kinetic energy theorem implies that the ball's kinetic energy must by greater at y = 0 than at y = 10 m. The same will be true for the ball's speed v.
(c) Calculate the net work W done on the ball during the ball's motion from y = 10 m to y = 0. Eq.(1) above gives an expression for W. However, dy is not given by Eq.(2) in the present case. Compare your value for W to that shown in the Data box.
(d) By the work-kinetic energy theorem, W should be equal to the change DKE in the ball's kinetic energy as the ball goes from y = 10 m to y = 0.
Thus,
W = -mg(0 - 10 m) = DKE = KEground - KE(0). (4)
Note that in the present case the left-hand side of Eq.(4) is positive. Therefore Eq.(4) implies that the kinetic energy of the ball is greater at ground level than at y = 10 m.
Use Eq.(4) and the value of KE(0) found earlier to calculate KEground. Use the applet to determine KEground 'experimentally'. Compare the experimental value of KEground to the calculated value.
(e) From KEground calculate the value of the speed vground when the ball reaches y = 0. Determine the value experimentally with the applet, and compare the calculated and experimental values.
(f) Repeat all of the above for a different mass, m = 400 g. Do you obtain a different value for h?
Explain your result theoretically by replacing KEground and KE(0) in Eq.(4) by (m/2)v2ground and (m/2)v2(0), respectively. Notice that the factor of m occurs on both sides of the equation and therefore can be canceled. The free fall motion of objects does not depend on their mass.