The following Activities are for the 'Work and Energy in Projectile Motion' 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 vertically upward with a given speed. 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) Determine KEtop and DKE.
(c) Compare the work W shown in the Data box to DKE.
(d) Calculate the force of gravity acting on the ball.
(e) From the work-kinetic-energy 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) What is the ball's kinetic energy KEtop when the ball reaches maximum elevation, and what is the change DKE in kinetic energy of the ball from the initial instant at t = 0 until the ball reaches maximum elevation?
(c) PLAY the motion, and PAUSE it when the ball is at its maximum elevation, as nearly as you can. You can STEP the ball up to this point and use a small step size when you get close. The Data box shows the ball's y-coordinate. Watch it as you approach the top. If you overshoot, REWIND the applet and PLAY/STEP to the top again.
The Data box shows the value of the work W done on the ball by the force of gravity acting on the ball. Since the force of gravity is the only force acting on the ball, W is equal to the net work done on the ball. Is the work W done while the ball is moving to the top equal to the value of DKE you determined above, as required by the work-kinetic energy theorem?
(d) The force of gravity acting on the ball is constant throughout the ball's motion. The applet will display it in green when you click the Force button. Calculate the force of gravity acting on the ball, and compare your result with that shown in the Data box.
The Data box displays the components Fx and Fy of the force. Note how the direction of the y-axis is defined in the applet and the corresponding sign of the y-component of the force of gravity.
(e) When the ball moves from its initial elevation y(0) = 10 m to its maximum elevation ymax = h, its displacement dy in the y-direction is equal to h - 10 m. You can display the ball's displacement vector (in blue) by clicking the Displacement button. Note that the displacement is opposite to the force of gravity, which means that the work done by the force is negative.
The net work W done on the ball during this displacement is equal to
W = Fydy = (-mg)dy = -mg(h - 10 m). (1)
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 = h.
Thus,
W = -mg(h - 10 m) = DKE. (2)
Note that in the present case both sides of Eq.(2) are negative.
Use Eq.(2) 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.(2) by (m/2)D(v2), where v is the ball's speed. 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.
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 vertically upward with a given speed from a given point above the ground. 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, and observe both the displacement of the ball and the
force acting on the ball.
(c) Calculate the net work W done on the ball 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 done on the ball during the ball's motion from y = 10 m to y = 0. Compare your value to that shown in the Data box.
(d) The net work W done on the ball during this displacement is equal to (-mg)dy = -mg(0 - 10 m). By the work-kinetic energy theorem, this should be equal to the change DKE in the ball's kinetic energy as the ball is going from y = 10 m to y = 0.
Thus,
W = -mg(0 - 10 m) = DKE = KEground - KE(0). (3)
Note that in the present case the left-hand side of Eq.(3) is positive. Therefore Eq.(3) implies that at ground level the kinetic energy is greater than at y = 10 m.
Use Eq.(3) and the value of KE(0) found earlier to calculate KEground. Then 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.(3) 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.