I. Selection of Step

We chose a 2.7 g ping-pong ball, horizontally placed and attached to a spring having k = 18 N/m and a compressed length of 6.5 cm. The ping pong ball will collide with another ping pong ball that is hanging by a 50-cm thread. The balls will clump together and swing up to hit the third ball at the top of a ramp. The ping pong balls reach a height of 0.35 m at the top of the swing; so the ramp height is 0.35 m higher than the beginning height of the ping pong balls.

II. Step A

a. Description

The Ball A was initially stationary, coupled to a compressed spring in a horizontal position. In addition, another Ball B was suspended from the ceiling by a thread, and the two balls were at the same level. For this condition, the height of these two balls is set to zero (h = 0). Because it was coupled to a compressed spring, the Ball A possessed elastic potential energy. Ball B, on the other hand, had no potential energy since h = 0. The compressed spring was released in the second phase, and Ball A collided with Ball B, transforming its stored potential energy into kinetic energy. Because the velocity was zero under the static condition, the momentum of both balls was zero.

b. Equations

In the previous step, the ball A is attached to a spring and is compressed prior collision with ball B. Under these conditions, the ball A only possesses elastic potential energy due to spring compression. The Equation 1 for ball A used in step A is given below.

Elastic potential energy stored in spring=1/2 k∆x_A^2 (1)

Where k is the spring constant and xA is the displacement of the spring. Equation 1 shows that the energy stored in the spring is directly proportional to the displacement of the spring from its initial position. When the ball A is released, it moves towards ball B and strikes with it. Before collision, the elastic potential energy stored in the spring is converted to kinetic energy of ball A, as the ball is now in motion. In this step, the energy stored in the spring is transferred to the ball A and causes the ball to move with velocity vA, until it strikes with ball B. The conservation of energy is observed during this step as the potential energy of the spring equals the kinetic energy of ball B. Kinetic energy of ball B is calculated using Equation 2.

Kinetic energy of ball A prior to collision=1/2 m_1 v_A^2 (1)

Where m1 is the mass of ball A and vA is the velocity of ball A just before collision. By solving Equations 1 and 2, it can be proved that energy is conserved while moving from previous step to the current step. It only changes from one form of energy to another form.

c. Calculation

Using the known values of k, x1, m1, and v1, Equations 1 and 2 are solved. The calculations are shown below.

Elastic potential energy stored in spring=1/2(18)〖(0.065)〗^2

Elastic potential energy stored in spring=0.038 J

Now, the kinetic energy of ball A is calculated using Equation 2.

Kinetic energy of ball A prior to collision=1/2(0.0027)〖(5.31)〗^2

Kinetic energy of ball A prior to collision=0.038 J

Law of conservation of energy is satisfied for this previous step as 0.038 J of elastic potential energy stored in compressed spring is completely converted to 0.038 J of kinetic energy of ball A.

Elastic potential energy of spring = Kinetic energy of ball

If we know the parameters of spring and its displacement, Equations 1 and 2 can be used to find out the velocity of object (in this case, ball A). After knowing the velocity of object, its location at any time can be computed using the definition of velocity.

Location or position of object=velocity x time

III. Step B

a. Description

In the current step, Ball A, gained momentum and collided with Ball B when released at a certain initial velocity. When two objects collide, the amount of momentum that is transferred is of significance. When two balls collided, the total momentum before and after the collision was equal to the total momentum of each ball considering there were no external forces involved. This is referred to as the law of conservation of momentum (Xu et al., 2020).

b. Equations

The current step is considered as the phase in which ball A strikes with ball B and sticks to it. Ball B, in step B, is suspended by a string and is at rest prior to collision. Since, only ball A is in motion prior to collision, thus it will have an initial momentum, as given by Equation 3.

Momentum before collision=Momentum of ball A=m_1 v_B (3)

As the ball A and B interact from the previous step to the current step, the law of conservation of momentum must hold. As soon as ball A strikes with ball B, both the balls move with a new velocity of vC. In current step, the kinetic energy of ball A converts to the combined kinetic energy of balls A and B. The final momentum after collision between balls A and B is given by Equation 4.

Momentum after collision=Total momentum of balls A and B=m_Total v_C (4)

Where mTotal is the combined mass of balls A and B and vC is the final velocity of balls A and B after collision. According to conservation of momentum:

Momentum before collision = Momentum after collision

c. Calculations

As we move from the previous step to the current step, the initial and final momentums are calculated using Equation 3 and 4. The final velocity of balls A and B after collision is measured to be 2.61 m/s.

Momentum before collision=0.0027*5.31=0.014 kgm/s Momentum after collision=0.0054*2.61=0.014 kgm/s

Conservation of momentum holds for our Rude Goldberg device as we move from previous step to current step.

Momentum before collision = Momentum after collision

IV. Step C

a. Description

The Ball A after colliding with ball B, was hooked to the Ball B and pushed the Ball B to a particular height. In the subsequent step, both balls regained their potential energy. Energy is converted from potential to kinetic energy in one phase and then back to potential energy in the next, yet the overall energy of the system is conserved according to the law of conservation of energy (Chernyavsky & Gapon, 2020).

b. Equations

As the system advances and moves from current step to the subsequent step, the energy transfer goes on. In this case, the kinetic energy of both the balls is converted to the gravitation potential energy of the balls A and B. As the balls collide in current step, because of momentum gained, they keep on moving with a velocity vC until they reach maximum height ‘h’. The kinetic energy of the system of balls is given by Equation 5.

Kinetic energy of balls A and B=1/2 m_Total v_C^2 (5)

As the balls move and reach a height ‘h’ above ground level, the kinetic energy is converted to potential energy and ultimately the balls come to a stop when vC becomes zero. The total gravitational potential energy in the subsequent step is given by Equation 6.

Total gravitational potential energy in final step=m_Total gh (6)

Where g is the acceleration due to gravity and h is maximum height attained by ball B. According to law of conservation of energy, the total kinetic energy in the current step must equal the total potential energy in the subsequent step.

Kinetic energy of balls A and B = Gravitational potential energy of balls A and B

c. Calculation

The height ‘h’ attained by ball B in subsequent step is measured to be 0.35 m. Equations 5 and 6 are solved to prove that conservation of energy holds after collision.

Kinetic energy of balls A and B=1/2(0.0054)〖(2.61)〗^2=0.018 J

Total gravitational potential energy in final step=0.0054*9.8*0.35=0.018 J

Hence, the system in the subsequent step observes the conservation of energy.

Kinetic energy of balls A and B = Gravitational potential energy of balls A and B

If we know the velocity of object before collision in current step, the velocity of the total system after collision can be obtained. Furthermore, using the velocity of the total system, the maximum height or location of the object in the subsequent step can be calculated using conservation of energy principle.

References

Chernyavsky, D., & Gapon, D. (2020). The relationship between the laws of conservation of energy and momentum for low speed impact of several bodies. Journal of Applied Nonlinear Dynamics, 9(2), 259-271.

Xu, W., Liu, Q., Koenig, K., Fritchman, J., Han, J., Pan, S., & Bao, L. (2020). Assessment of knowledge integration in student learning of momentum. Physical Review Physics Education Research, 16(1), 010130. https://doi.org/10.1103/PhysRevPhysEducRes.16.010130

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