Elastic/Inelastic Collisions

Transcription

1 Lab #6 Collisions page 1 Elastic/Inelastic Collisions Reading: Giambatista, Richardson, and Richardson Chapter 6 (6.7), Chapter 7 ( , 7.7). Summary: In proton cancer therapy, positively charged particles are made to move very fast (next semester you will learn how this is done) and then directed at cancerous tumors. There are several types of interactions the particles can have with tumor cells and, in this lab, you will investigate two of the interactions. The other interactions you will study next semester when you learn about charged particles. The two specific interactions you will explore in this lab are inelastic and elastic collisions. Of the two, inelastic collisions are the more desirable in proton treatments because almost all of the proton energy is absorbed by the tumor, thereby destroying it quickly. However, elastic collisions can also damage tumors but in a different way. In this lab, you will simulate the protons with a cart, while the tumor cells will be simulated by another cart with a 500g weight on it, since the tumor cells are more massive than the protons. You will then push the proton towards the tumor cells, where they will collide first elastically and then inelastically. A retractable spring and piece of Velcro on the carts will allow you to select the type of collision. During the actual cancer therapy, the doctor can only measure the speeds of the incoming proton and the particles after the collision has occurred. From that information, it is possible to determine which type of collision occurred and thus how efficient the therapy was, i.e. if the collision was mostly inelastic then the therapy was more efficient. In this lab, you will learn how the doctor determines which type of collisions occurred from the measured speeds. To do this, you will need to derive the velocity relations appropriate for one-dimensional elastic and inelastic collisions. The Pre-Lab Analysis helps you through this using conservation of momentum and energy. Pre-Lab Analysis In the lab, all the collisions you will perform will involve the two carts rolling on an aluminum track. Since the track constrains the cart motion to one dimension, all the formulas you will derive in this Pre-Lab will be for one-dimensional motion. For both types of collisions, the final velocities are related to the initial velocities by the ratio of the masses. However, the ratio is different depending on whether the collision is elastic or inelastic and this is how the doctor can tell from the measured speeds before and after the collision what type of collision must have occurred. The first two problems in this Pre-Lab walk you through the derivation of the velocity relations for the two types of collisions. In all iinneellaassttiicc ccoolllliissiioonnss, only moomeennttuum is ccoonnsseerrvveedd. In the proton cancer therapy, an example of a perfectly inelastic collision is when the proton hits a tumor cell, sticks to it, and both move away from the collision site with the same velocity. Another somewhat more familiar example of a perfectly inelastic collision is when a fast moving car runs into a second one stopped at a traffic light. After the collision both cars stick together and usually end up skidding away from the collision point. Because they are stuck together, they each skid with same velocity as the other. 1. Perfectly inelastic collisions: let m1 be the mass of the proton and m2 the mass of the cell into which the proton collides. The proton s initial velocity is v1i, the cell s initial velocity is v2i = 0, and the final velocity of the proton and cell stuck together is vf. Using conservation of momentum, show that: [4 pts] 1

2 Lab #6 Collisions page 2 v f = + m 2 v 1i (proton and cell stick together perfectly inelastic) In all eellaassttiicc ccoolllliissiioonnss, moomeennttuum aanndd eenneerrggyy are ccoonnsseerrvveedd. In the proton cancer therapy, an example of a perfectly elastic collision is when the proton hits a tumor cell, bounces off it, and both move away from the collision site with different velocities. Another somewhat more familiar example of a perfectly elastic collision is when two people on rollerblades collide. After the collision, both people end up flying apart at different velocities (usually). 2. Perfectly elastic collisions: let m1 be the mass of the proton and m2 be the mass of the tumor cell with which the proton collides. The proton s initial velocity is v1i, the stationary cell s initial velocity is v2i = 0, the final velocity of the proton is v1f and of the tumor cell is v2f. a.) Write the energy and momentum conservation equations for this collision. [4 pts] 2 b.) From the momentum equation show that [2 pts] 3 m 2 ( v 1i - v 1f ) = v 2f c.) From the energy equation show that [2 pts] 4 m 2 v 2-2 ( 1i ) v1f = v 2 2f 2 d.) Square both sides of (b) and substitute into (c) for v 2f and show that [4 pts] 5 m 2 = v 1i + v 1f v 1i - v 1f Hint: you will need to use the mathematical identity (a 2 -b 2 )=(a+b)(a-b) e.) Show that this last equation can be rewritten as [3 pts] 6 Ê v 1f = - m 2 ˆ Á v Ë + m 2 1i (carts collided elastically) f.) If the (stationary) cell was replaced with a solid wall (i.e. m2æ ), show that this last equation reduces to v1f = v1i. (Hint: first multiply the top and bottom of the above equation by 1 m2 ; next simplify and then take the limit as m2 goes to infinite.) [4 pts] 7 Thus, one of the easiest ways to tell if a collision is elastic or inelastic is to plot v1f versus v1i. If the slope of the resulting curve is the ratio of the difference to the sum of the masses (as in Question 2e), then the collision was elastic. If it is not, then the collision must have been inelastic. You will need to use this in the lab and in the next question.

3 Lab #6 Collisions page 3 3. The two tables to the right consist of initial and final velocities for a cart that collided with a stationary second cart. a.) In the first table the two carts fly apart after the collision. Plot v1f versus v1i and fit a linear Trendline to it. [8 pts] 8 b.) The mass of the moving cart (before the collision) was 600 grams and the mass of the stationary cart was 500 g. From these masses and the slope of the Trendline in part (a), determine if the collision was elastic or inelastic. Explain your answer. To obtain full credit for your answer, you must show all your work (for example, any plots you make should be printed and included in your answer). [8 pts] 9 c.) In the second table the two carts stick together after the collision. Is this an elastic or inelastic collision? Justify your answer. [4 pts] 10 d.) Plot the final velocity of the two carts stuck together (y-axis) versus the starting velocity of the initially moving cart (x-axis). Fit a linear Trendline to your curve. [9 pts] 11 V1initial (m/s) V1final (m/s) Table 1: data for parts (a), (b). v1initial (m/s) vfinal (m/s) Table 2: data for Question 3(c) (e). e.) If the moving cart 1 weighed 600 grams, what was the weight of cart 2 (the initially stationary cart)? (Hint: use the slope of the line in 3d above with one of the equations from Question 1 or 2.) [5 pts] Summarize and outline the lab following the format posted on the Electronic Reserves course web page. [20 pts] 13 Equipment to be used in this lab: proton (cart) with an extendable spring = Cart #1. tumor cell (cart with attached angle bracket = Cart #2 and 500 g weight on it). m Store all carts on their sides - their wheels are delicate. motion sensor connected to Port 2. m 2 aluminum stop bars: one at each end of the track. m Bubble level. Experimental Procedure: 1. Computer Setup q Check that the motion sensor is connected to port 2. q Check that the force sensor is connected to DIN 1 q Check that the switch on the force sensor is set to ±50N. q Click on the Lab 6 Collisions icon. Make sure both distance vs time and velocity vs time graphs are present. q Check that Experiment Length =2 sec and the Sampling Speed = 15 samples/sec.

4 Lab #6 Collisions page 4 q Check that the track is roughly level using the bubble level. If it is not, notify your TA. 2. Cart Characterization q Weigh both the proton and the tumor cell. Make sure the unweighted triple-beam scale balances at zero before weighing the particles. Weigh the tumor cell without the 500g mass on it. The 500 g mass is very accurate, so you can assume its mass is precisely 500g. [4 pts] 14 m q Calculate 1 and circle your answer for later reference. [4 pts] q Calculate 3. Elastic Collision? a.) Data Collection Ê - m 2 ˆ Á and circle your answer for later reference. [4 pts] Ë + m 2 16 q Place the proton on the track with its spring pointing towards the tumor cell. The tumor cell should be kept stationary before the collision. q Separate the proton and tumor cell by about 30 cm. Space them on the track so the collision point is more than 0.4 m from the motion sensor, or it won t detect the collision. q Click on Collect. Immediately after the initial two sensor clicks, push gently and let go of the proton. If necessary, after the collision, stop the tumor cell before it hits the bar at the far end of the track. b.) Data Analysis i.) Determining v1i, v1f q Click the X=? icon and use the pointer to read directly v1i and v1f immediately before and after the collision. q Record these values in a data table in your lab notebook. Don t forget to label the data table, all the columns in the table, and include units on all columns. q Repeat your measurements two more times using different initial proton speeds (push the cart with a different force each time) and record the v1i and v1f for each trial in your notebook. [12 pts] 17 q Print one representative plot each of distance vs time and velocity vs time. Label which trial # each plot corresponds to. [4 pts] 18 ii.) Is collision elastic? q In Excel enter the data from your lab notebook. q Plot the initial (on the x axis) and final velocities of the proton and fit a linear Trendline to the data. Print your results with the Trendline and its equation showing. [10 pts) 19 q Compare the slope of this Trendline line to the mass ratios you found in Part 2 (find the % differences). [6 pts) 20 q From your comparisons, determine if this collision was elastic or not. Explain your reasoning. Answer in a full sentence. (Hint: look at the reasoning you used in Prelab

5 Lab #6 Collisions page 5 4. Inelastic Collisions Question 3.) [7 pts] 21 Because of experimental errors (either human or equipment), your measurements will never give as precise results as the theory. If your percent difference is less than ~15%, then you can conclude that this was mostly an (in)elastic collision. a.) Data Collection q Retract and latch the spring on the proton cart. (Push up on the black knob as you retract the spring to lock it.) q Point the velcroed side of the tumor cell (the one closest to the sensor) towards the other proton. Check that the velcro holds the proton to the tumor cell when they collide. q Click on Collect and gently shove the proton towards the tumor cell. If they don t stick when they collide, redo the collision and retake the data. b.) Data Analysis i.) Determining v1i, v1f ii.) q In LoggerPro, use the slope of the distance vs time graph to determine the proton s velocity just before and just after the collision (i.e. v1i and v1f). Record these slopes in a new, appropriately labeled, data table in your lab notebook. [8 pts] 22 Instructions for finding slope in LoggerPro: Select a small section of the curve (just before the collision) over which the velocity is reasonably constant Click on the R= icon this fits a line to the selected portion of the curve and gives the equation of that line. Repeat this procedure to find the velocity just after the collision. q Check at least one of these velocity values by reading the velocity directly using the X= icon. Since it is hard to tell the precise point at which the collision begins and ends, the slope method should give a better idea of the average velocity than the X= icon. Just make sure the velocities found using the two different methods are not wildly different from each other. q Repeat this collision experiment two more times with different initial proton speeds. Determine the velocities using the slope method described above. Record all values in your data table. [4 pts] 23 q Print a representative plot. Label which trial # the plot corresponds to. [2 pts] 24 Is collision inelastic? q In Excel, enter the data from your notebook. Plot the initial (on the x axis) and final velocities of the proton and fit a linear Trendline to the data. Print your results with the Trendline and its equation showing. [10 pts) 25 q Compare the slope of this Trendline line to the mass ratios you found in Part 2 (find the % differences). [6 pts) 26

6 Lab #6 Collisions page 6 Extra Credit q From your comparisons, determine if this collision was elastic or inelastic. Explain your reasoning. Answer in a full sentence. (Hint: look at the reasoning you used in Prelab Question 3.) [7 pts] Energy Loss in Inelastic Collisions q Choose one trial from Part 4 above. q From the initial velocity you experimentally measured and the measured proton and tumor cell masses, find the total initial kinetic energy of the two- particle system. [4 pts] 28 q From the final measured velocity of the proton stuck to the tumor cell, find the final kinetic energy of the two- particle system. [4 pts] 29 q Record all the kinetic energies in an appropriately labeled table in your notebook. Don t forget to include units on all columns and which Trial # you used in your calculations. [9 pts] 30. q Calculate the percentage of the initial kinetic energy lost during the collision. [3 pts] 31 q Where did this energy go? (Again, no credit if your answer is not in a full sentence.) [3 pts] 32.