collisions in one dimension lab report

Experiment 6: Collisions in One Dimension Irene Zou General Physics I: Laboratory Fall 2023, Section 020 Instructor: V. Mehta Partner(s): Tasneem Abdalla Thursday, November 9th, 2023 Due: Thursday, November 16th, 2023

Zou 2 Objective: The purpose of this experiment is to understand and look into conservation of energy, as well as momentum, in one dimensional, two body system collisions. Two different one dimensional collisions were performed, perfectly inelastic and elastic. Description/Procedure: For elastic collisions, one glider will stay at rest on the air track while the other glider strikes the glider that is at rest. The glider striking the other glider at rest can be of the same mass, smaller, or larger. There were two red gliders, weighing at 305.3g and 304.6 g, and one blue glider that weighed 451.0g. For the same masses, both of the red gliders were used. When the striking glider had a smaller mass than the one at rest, the red glider was used to strike the blue glider. When the striking glider had a larger mass, the blue glider was used to strike the red glider. Two photogates are used to measure the velocities of the gliders before and after the collision, along with the Capstone program. For setting up the photogate in Capstone, the length of the index card that is paper clipped to the glider was set as the flag width. The length of our index card was 7.5cm, so for flag width, we put 0.075m. Inelastic collisions used the same setup and materials. The only difference was that for inelastic, we inserted the inelastic collision needs to one side of the glider and the wax receptacle into the other. Five trials were performed for each variation of masses for both collisions. The velocities collected were then used to calculate momentum and kinetic energy. Theory: Collisions only produce internal forces, so momentum is the same before and after a collision, hence momentum is conserved. If all components of force ( F ) equals 0, then all components of momentum are conserved. A collision is considered elastic if the combined kinetic energy of the two objects is the same before and after the collision. If kinetic energy is less after the collisions, then it is an inelastic collision. Inelastic collisions are when two objects stick together causing maximum kinetic energy to be lost. For elastic collisions, energy is given by: 1 2 ? 1 𝑣 1 2 = 1 2 ? 1 𝑣' 1 2 + 1 2 ? 2 𝑣' 2 2 where the gliders are referred to by the subscripts 1 and 2. Unprimed is for velocity before the collision, 𝑣 is for velocity after the collision, and is for the mass of the gliders. Before the collisions, glider 2 𝑣' ? will be at rest and the direction of motion for glider 1 will be taken as positive. Momentum of elastic collisions is given as: ? 1 𝑣 1 = ? 1 𝑣' 1 + ? 2 𝑣' 2 For both of these equations in elastic collisions, is assumed as known. We can rearrange these two 𝑣 1 equations to solve for and as seen below: 𝑣' 1 𝑣' 2 𝑣' 1 = ( ? 1 −? 2 ? 1 +? 2 )𝑣 1

Zou 4 ? 1 < ? 2 4 0.650 0.000 0.132 0.507 0.0644 0.0606 0.198 0.269 ? 1 < ? 2 5 0.777 0.000 0.133 0.589 0.0921 0.0809 0.237 0.306 ? 1 > ? 2 1 0.387 0.000 0.151 0.764 0.0338 0.0942 0.174 0.301 ? 1 > ? 2 2 0.420 0.000 0.116 0.701 0.0340 0.0780 0.189 0.266 ? 1 > ? 2 3 0.207 0.000 0.105 0.651 0.010 0.0671 0.0934 0.246 ? 1 > ? 2 4 0.360 0.000 0.145 0.825 0.0292 0.109 0.162 0.317 ? 1 > ? 2 5 0.261 0.000 0.128 0.761 0.0154 0.0920 0.118 0.290 Table 4. Inelastic Collision Mass Trial 𝑣 1 (m/s) 𝑣 1 + 𝑣 2 (m/s) 𝑣' (m/s) 𝑃 (kg m/s) · 𝑃' (kg m/s) · ∆𝑃 (kg m/s) · 𝐾? 𝑖 (J) 𝐾? ? (J) ∆𝐾? (J) ? 1 = ? 2 1 0.321 0.316 0.161 0.0979 0.0982 3. 00 × 10 −4 0.0157 0.00791 0.00779 ? 1 = ? 2 2 0.315 0.310 0.158 0.0961 0.0964 3. 00 × 10 −4 0.0151 0.00761 0.00749 ? 1 = ? 2 3 0.330 0.327 0.165 0.101 0.101 0.00 0.0166 0.00830 0.0083 ? 1 = ? 2 4 0.386 0.383 0.193 0.118 0.118 0.00 0.0277 0.0114 0.0163 ? 1 = ? 2 5 0.339 0.336 0.170 0.103 0.104 0.001 0.0175 0.00881 0.00869 ? 1 < ? 2 1 0.236 0.235 0.0952 0.0720 0.0720 0.00 0.00849 0.00343 0.00506 ? 1 < ? 2 2 0.285 0.283 0.115 0.0869 0.0869 0.00 0.0124 0.00500 0.00740 ? 1 < ? 2 3 0.258 0.255 0.104 0.0787 0.0786 1. 00 × 10 −4 0.0102 0.00409 0.00611 ? 1 < ? 2 4 0.284 0.291 0.115 0.0866 0.0869 3. 00 × 10 −4 0.0123 0.00500 0.00730 ? 1 < ? 2 5 0.285 0.286 0.115 0.0869 0.0869 0.00 0.0124 0.00500 0.00740 ? 1 > ? 2 1 0.454 0.437 0.271 0.205 0.205 0.00 0.0465 0.0278 0.0187 ? 1 > ? 2 2 0.362 0.357 0.216 0.163 0.163 0.00 0.030 0.0176 0.0124

Zou 5 ? 1 > ? 2 3 0.373 0.323 0.223 0.168 0.169 0.001 0.0314 0.0188 0.0126 ? 1 > ? 2 4 0.550 0.487 0.328 0.248 0.248 0.00 0.0682 0.0407 0.0275 ? 1 > ? 2 5 0.366 0.361 0.218 0.165 0.165 0.00 0.0302 0.0180 0.0122 ? 1 (??𝑎?? ??𝑖???) = 305. 3? (0. 305??) ? 1 (?𝑎??? ??𝑖???) = 451. 0? (0. 451??) ? 2 (??𝑎?? ??𝑖???) = 304. 6? (0. 305??) ? 2 (?𝑎??? ??𝑖???) = 451. 0? (0. 451??) elastic collision equations: 𝑣 1 ' = ? 1 −? 2 ? 1 +? 2 𝑣 1 𝑣 2 ' = 2? 1 ? 1 +? 2 𝐾? 𝑖 = 1 2 ? 1 (𝑣 1 ) 2 𝐾? ? = 1 2 ? 1 (𝑣 1 ') 2 + 1 2 ? 2 (𝑣 2 ') 2 𝑃 𝑖 = ? 1 𝑣 1 𝑃 ? = ? 1 𝑣 1 ' + ? 2 𝑣 2 ' inelastic collision equations: 𝑣' 1 = 𝑣' 2 = ? 1 ? 1 +? 2 𝑣 1 𝐾? 𝑖 = 1 2 ? 1 (𝑣 1 ) 2 𝐾? ? = 1 2 ? 1 (𝑣 1 ') 2 + 1 2 ? 2 (𝑣 2 ') 2 𝑃 = ? 1 𝑣 1 𝑃' = (? 1 + ? 2 )𝑣' 1 Error Analysis: A source of error in this experiment is random error. Since there were two photogates, you had to pay attention to which photogate was recording which glider. This could have caused us to record and collect the wrong information. The photogates also could not have been spaced out enough or too close to each other as we were working with limited space and air track. The air track was also not leveled before the start of the experiment, causing errors. The air track is also not entirely frictionless, causing the first run of inelastic collisions to be faster than the other trials ran after. Friction causes kinetic energy to leave the system. In elastic collisions, there could have been energy lost due to the sound the gliders make when they collide. 𝑣 (??𝑎??𝑖? ?????) = 0.836−0.787 0.787 × 100 = 6. 23% 𝑣 (𝑖???𝑎??𝑖? ?????) = 0.338−0.334 0.334 × 100 = 1. 20%

Zou 7 In elastic collisions, the kinetic energy should be the same before and after the collision. However, in my data this is not the case. A general trend for kinetic energy during elastic collisions is that final kinetic energy is lower than the initial kinetic energy. I expect a bit of kinetic energy to be lost in the collision because the air track is not perfectly frictionless. Loss of kinetic energy is probably due to this friction of the air track, as this was not performed in a vacuum. Based on the data table, we can also see that momentum is also not conserved like it should be. The values for momentum should be the same for before and after collisions. 6. (3.4 Elastic Collisions) What is occurring during the collision? What happens to the energy during the collision? Explain in the case of spring bumpers. If you have used disk magnets, how would they influence the experiment? Explain. In an elastic collision, the two gliders are colliding against each other. One glider is stationary and the other glider hits the stationary glider, causing the stationary glider to move forward and the striking glider to bounce back and move in the opposite direction. Ideally, the gliders move in opposite directions at a constant velocity as before the collision. Therefore, energy during the collision should be conserved, so the same before and after the collision. In spring bumpers, friction plays a role in why energy would be lost and causing the energy of the system to be smaller. If disk magnets were used, it could reduce the force of friction causing less energy to be lost. 7. (3.5 Perfectly Inelastic Collisions) Analyze your data in the same way. Use the equation for a perfectly inelastic collision to verify that KE is lost. What is occurring to the energy? How does the energy get dissipated? In a perfectly inelastic collision, the maximum total kinetic energy is lost. This is because after colliding, the gliders stick together and move as one glider causing the final velocity of the gliders to decrease (move more slowly). 𝐾?' = 1 2 (? 1 + ? 2 )𝑣 2 = 1 2 (2? 1 )𝑣 2 = ? 1 𝑣 2 𝐾? = 1 2 ? 1 𝑣 2 ? 1 𝑣 2 > 1 2 ? 1 𝑣 2 𝐾?' > 𝐾? As seen in the data above, the kinetic energy after the collision is significantly lower than the kinetic energy before the collision. This coincides with the theory above kinetic energy not being conserved in inelastic collisions. However, momentum is relatively conserved as seen by the similar values in momentum before and after collision. 8. (Comment) When the gliders collide do you hear a sound? Could this contribute to the energy and momentum balance in your experiment? How does this compare with the kinetic energy of the gliders?

Zou 8 When the gliders collide, we do hear a sound. This could have contributed to the energy and momentum balance in your experiment because this sound causes some energy to be lost. However, it is not the only cause of energy to be lost. The kinetic energy transferred between the gliders is higher than the kinetic energy of the sound of the collision. This is because sound energy travels through the air that transfers a small amount of energy whereas gliders transfer energy between each other and each glider has a larger mass.