The following Activities are for the Work and Kinetic Energy 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 verify the Work-Kinetic Energy Theorem for parabolic motion (constant net force).

Exercise 1. Suppose a particle is moving subject to a constant net downward force . Figure 1 below shows part of the parabolic path of the particle, and it shows the particle in two positions that are 2.0 s apart and data corresponding to this time interval.

Figure 1

Using the data from Figure 1, perform the following calculations to verify the Work-Kinetic Energy Theorem for this 2.0-s time interval.

• Calculate the work done on the particle by the net force as the particle undergoes the displacement D. Note that, since the net force is constant, = _av.
  
  The work done by a constant force is equal to D. Evaluate this scalar product, and check if you obtain the result listed in Figure 1.
  
  Compare the value of D to that of W. The two values should be exactly equal in the case of a constant net force. However, you will notice that the two values listed in Figure 1 differ in the second decimal. This discrepancy is the result of round-off error due to giving the net force to only two significant figures.
  
  W denotes the exact value of the work done by the net force in the given time interval while _avD in general only approximates W. However, for a constant net force, _avD is equal to W.
  
  
• From the value of the velocity at the beginning of the 2-s time interval, _old, calculate the speed at the beginning of the interval, v_old, and the kinetic energy at the beginning of the interval, K_old. Compare your results with the values in Figure 1.
  
  Do the same with the value of the velocity at the end of the 2-s interval, _new.
  
  Then verify the Work-Kinetic Energy Theorem by demonstrating that
  W = K_new - K_old.
  
• Verify that the values of _old, _new, and _av from Figure 1 are consistent by veryfing the equation
  
  _new = _old + (_av /m) Dt.
  For constant net force = _av and therefore constant acceleration _av /m, this equation is exact, no matter how large the time interval Dt is. For variable net force and acceleration, it becomes exact only in the limit of vanishing Dt.

Exercise 2. Use the applet to create a parabolic motion of your own by applying a constant net force. You may want to choose a net force that is not downward, but in some other direction. For this motion, carry out the same kinds of calculations as in Exercise 1 for one or more time steps.

Activity 2. The purpose of Activity 2 is to investigate the Work-Kinetic Energy Theorem for a motion with variable net force.

Exercise 1. Use the applet to create a motion with variable net force.

REWIND the applet and set the time step to 2.0 s. STEP FORWARD through the motion, and stop at some 2.0-s time interval. Using the values in the data box, verify that the equation of the Work-Kinetic Energy Theorem,

is satisfied.

By stepping either forward or backward to other 2.0-s intervals, you can check that the preceding equation is satisfied for all other such intervals as well. This means that the equation is also satisfied for larger intervals, e.g., 10-s intervals, because the work done and the change in kinetic energy in such an interval is the sum of the work and change in kinetic energy, respectively, in the constituent 2.0-s intervals.

The question remains: "How can we calculate the work W listed in the applet's data box using the basic definition of work?" The purpose of the following Exercise 2 is to answer this question.

Exercise 2. Continuing from Exercise 1, REWIND the applet.

STEP FORWARD through the motion. For some 2.0-s time interval, calculate the approximate work _avD done. Compare this value to that of the exact work W listed in the data box. If the force is varying during your time interval, the two are in general not going to be equal.

The approximate work _avD approaches the true work W as the size of the time interval decreases. Investigate this as follows.

Suppose t = t₁ is the instant of time at the beginning of the 2.0-s interval. REWIND the applet, and reduce the size of the time step from 2.0 s to 1.0 s. STEP FORWARD through the motion until t_old = t₁, i.e., until you are at the beginning of the former 2.0-s interval. Now, t_new = t₁ + 1.0 s. For the 1.0-s interval from t₁ to t₁ + 1.0 s, i.e., for the first half of the former 2.0-s interval, compare the values of W and _avD displayed in the data box. You can verify the latter by calculation, if you like.

Is the percentage difference between W and _avD smaller for the 1.0-s interval than for the 2.0-s interval? I.e., is |W - _avD| / W smaller for the 1.0-s interval than for the 2.0-s interval?

Cut the interval in half again, to 0.5 s, and repeat the comparison between W and _avD, and so on for even smaller intervals. (The smallest interval allowed by the applet is 0.2 s.)

What do you conclude?

Exercise 3. One can learn from Exercise 2 that approximating W by _avD involves an error, but an error that vanishes as the interval size vanishes.

This leads to the following prescription for calculating W for some finite time interval. Divide the interval into many smaller sub intervals, calculate _avD for each of the smaller intervals, and sum all these small approximate amounts of work to get an approximate value for W. Repeat this by dividing the given time interval into even more and even smaller subintervals, calculating and summing the many small amounts _avD, and so on for even more and even smaller intervals. The resulting sequence of approximate values for W converges to the exact value of W as the number of subintervals goes to infinity and the size of each subinterval goes to zero.

Suppose one divides a given 10-s time interval into one million time intervals of 1/100000 s each, and calculates _avD for each of these tiny intervals. Then the sum of a million tiny amounts _avD is going to be very close to the true value of W for the 10-s interval.

Computers can carry out calculations like this, called numerical integration, quickly. (The word "integration" means summation.)

Activity 3. The purpose of this Activity is to investigate the Work-Kinetic Energy Theorem qualitatively.

Exercise 1. Work can be either positive or negative, depending on whether the force doing the work forms an angle less than or greater than 90^o with the displacement. According to the work-kinetic energy theorem, the kinetic energy increases during a time interval for which the work is positive and decreases during a time interval for which the work is negative.

RESET the applet. Select the Force button. Generate a motion by means of the Force dial. Observe the red kinetic energy bar in relation to the angle between the black force vector and the magenta velocity vector.

Does the kinetic energy bar increase in height when the angle between the force and velocity is less than 90^o, i.e., when the force generally speaking is in the "forward" direction, and decrease in height when the angle between the force and velocity is greater than 90^o, i.e., when the force generally speaking is in the "backward" direction?

Activity 4. The purpose of this Activity is to repeat Activities 1 to 3 for motion along a straight line.

Exercise 1. RESET the applet. Set the applet to the 1D mode.

Repeat any of the exercises in Activities 1 to 3 using one-dimensional motions.