Momentum Can Be Calculated with the Following Definition

Collisions

When two objects collide, they exert forces on one another. If we know the exact forces and time of contact, then we can use Newton’s second law and kinematics equations to predict the subsequent motion of the objects. But the exact forces exerted can be extremely difficult to measure, and they often vary over the course of the collision, so a different method of analysis is preferred. The preferred analysis uses the law of conservation of momentum.

Momentum can be calculated with the following definition:

[1] p = mv p = momentum m = mass v = velocity

Newton’s second law can be rewritten in terms of momentum as follows:

[2] ΣF = ma (Newton’s second law) [3] ΣF = m(Δv/Δt) (definition of acceleration) [4] ΣF = Δ(mv)/Δt (assumption of constant mass) [5] ΣF = Δp/Δt (definition of momentum)

The second law can be stated as “the net force is the rate of change in momentum.” If we analyze a pair of objects with no significant net external force, then we can use equation [5] and Newton’s third law to prove conservation of momentum.

[6] F21 = Δp1/Δt (equation [5] applied to object 1) [7] F12 = Δp2/Δt (equation [5] applied to object 2) [8] F21 = -F12 (Newton’s third law) [9] F21 + F12 = 0 [10] Δp1/Δt + Δp2/Δt = 0 (equations [6] and [7] in equation [9]) [11] Δp1 + Δp2 = 0 (multiply by Δt) [12] (p1f - p1i) + (p2f - p2i) = 0 (definition of Δ) [13] p1i + p2i = p1f + p2f

For a system with no significant net external force, the total momentum before an event equals the total momentum after an event. This is called “conservation of momentum.” This analysis can be extended to more than two objects. In this lab, we will test the motion of two objects in one dimension. Conservation of momentum in one dimension is written as follows:

[14] p1ix + p2ix = p1fx + p2fx [15] m1v1ix + m2v2ix = m1v1fx + m2v2fx If two objects collide without significant transformations of energy to sound, thermal energy, or permanent deformation, then kinetic energy will be conserved. This special type of collision is called “elastic.”

[16] K1i + K2i = K1f + K2f 2 2 2 2 [17] ½m1v1ix + ½m2v2ix = ½m1v1fx + ½m2v2fx

If the collision is not elastic, then the expectation is that the total kinetic energy after the collision is less than the total kinetic energy before the collision.

If two objects stick together after a collision, then the collision is called “completely inelastic.” The conservation of momentum equation can be rewritten for this special case as follows:

[18] m1v1i + m2v2i = m1vf + m2vf [19] m1v1i + m2v2i = (m1 + m2)vf

Physics is fun!

Experimental Procedures

Error Estimation for Velocity

1) Obtain a single glider, measure its length, and place it at one end of the air track. 2) Place one photogate near the glider and place a second photogate near the end of the air track. 3) Turn on the air track and photogates. 4) Send the glider through the photogates. 5) Observe the time measurement on each photogate. 6) Calculate the velocity of the glider at each photogate using distance/time. 7) Calculate the difference in the velocity. This value can be used as the uncertainty in subsequent velocity calculations.

Collisions

8) Obtain two gliders. 9) Measure the mass and length of each glider. 10) Place the gliders on the air track and turn on the air track. One possible configuration is illustrated above. If you choose to have one of the gliders be a stationary target, it will need to be placed between the photogates. 11) Configure two timers to the correct height so that they measure the time it takes a glider to pass. The timers should be set to “gate” and “memory on”. 12) Shove one or both of the gliders so that they collide. If they collide with their Velcro sides, then it will be an inelastic collision. If the gliders collide with their bumper sides, then it might be an elastic collision. 13) Immediately record a description of the collision. For example, you might say something like, “A yellow glider was initially stationary in the middle of the track. We shoved a red glider to the right. It collided with the yellow glider and bounced off, moving to the left. The yellow glider moved to the right after the collision. 14) Take time readings from the timers. The display on the timer will be the time it took a glider to pass the first time. If a glider passed a photogate more than once, then flip the memory switch to “read” and it will give you the total elapsed time. You will need to subtract to obtain the time for a glider to pass the second time. 15) Calculate (or infer a zero if stationary) the velocity of each glider before and after the collision. You can use distance over time for magnitude of the velocity because it is not accelerating. You will need to decide on a coordinate system and indicate positive or negative for each velocity. Immediately record this information. If you do not do this, you might get a zero on your report and be required to repeat the experiment. 16) Use equation [15] to calculate if momentum was conserved in the collision. Be sure to use positive or negative velocity as needed in your calculations to indicate direction. The expectation is that momentum will be conserved in any collision. Note: you may use equation [19] instead if this is an inelastic collision. 17) Use equation [17] to calculate if kinetic energy was conserved in the collision. The expectation is that kinetic energy will be conserved (the same total initial and final) for an elastic collision and transformed (initial greater than final) for any other type of collision. 18) Repeat the experiment using different initial velocities, different type of collision, or different mass of glider. Complete a total of four trials.

Note: You will have four calculations of total initial momentum, four calculations of total final momentum, four comparisons of total initial momentum to total final momentum, four calculations of total initial kinetic energy, four calculations of total final kinetic energy, and four comparisons of total initial kinetic energy to total final kinetic energy. Because you are doing highly repetitive calculations, it is recommended that you use a spreadsheet. Note on units: You do not need to worry about exclusively using SI units in this experiment as long as you consistently use the same system of units. You are welcome to use CGS (centimeter, gram, second) units, for example. The CGS unit of momentum is the g*cm/s. The CGS unit of energy is the “erg” which is a g*cm2/s2.