Conservation of Momentum in Two Dimensions

Transcription

1 Conseration of Momentum in Two Dimensions Name Section Linear momentum p is defined as the product of the mass of an object and its elocity. If there is no (or negligible) external force in a collision, then momentum is consered. Theory Consider the two-dimensional (glancing) collision shown below. Here, mass m 1 traels to the right with elocity 1o and strikes mass m 2 initially at rest. After the collision, m 1 continues to the right with elocity 1f at an angle θ from the direction of its initial elocity (which we can coneniently make the x-axis of our coordinate system). m 2 acquires elocity 2f and traels at an angle ϕ from this direction. Momentum is a ector quantity; therefore the analysis of momentum inoles ectors. 1 1o p + o 2 2o = p f = 1 1 f f (1) Howeer, this analysis can be simplified by analyzing the momentum in each dimension separately. 1 1xo 1 1yo 2 2xo 2 2 yo = = 1 1xf 1 1yf 2 2xf 2 2 yf (2) With the collision defined as such, there are two things worth noting. First, m 2 is initially at rest, so it has no initial x or y momentum. Second, m 1 is initially traeling along the x-axis, so the initial momentum in the y dimension is zero. Thus, the final total momentum in this dimension will be zero (een though the indiidual momenta are not zero). Thus, the analysis of this collision reduces to 1 1xo = 1 1xf 0 = 1 1yf 2 2xf 2 2 yf (3) The collision in this experiment is between 2 small steel spheres. One is released from the top of a cured metal track. At the bottom of the track it collides with a second, stationary sphere. Both spheres are projected horizontally from here and fall onto newsprint taped to the floor below. Since there is no horizontal force on the spheres, they moe in this direction with constant elocity. If a sphere traels distance s across the newsprint, then its elocity is Su07 Page 1 of 6

2 where t its trael time. But, each sphere is in freefall once projected, so that where d is the freefall distance. s = (4) t 2d t = (5) g Apparatus Cured metal track, Clamp, Steel spheres, Triple-beam balance, Newsprint, Carbon paper, Tape, Plumb bob, Meterstick, Protractor. Procedure 1. Designate one of the spheres as m 1 and the other as m 2; measure their masses and record these alues in Table 1. Be sure to remember which sphere is which, as the masses might be different. 2. Measure the freefall distance d from the bottom of the track to the floor. Record this alue and calculate the free-fall time. 3. The sheets of newsprint are 2ft x 3ft; tape two together to make a larger 3ft x 4ft sheet. Place this on the floor such that it extends from under the bottom of the track out away from the table. 4. Set up the collision. Place m 2 on top of the target screw at the end of the Plexiglas base and adjust it so that the spheres will be at the same height when they collide. Also, the spheres should not oerlap by more than half you want a glancing collision. Adjust the track so this is the case (it can be moed left or right). Both situations are shown below. 5. Release m 1 from the top of the track and let it collide with m 2 sitting on the target screw. If both spheres do not land on the newsprint, then adjust its location so that they will the next time (a portion of the newsprint needs to remain under the target screw). Also, if both spheres did not strike the floor at the same time (listen), then the height of the target screw will need to be readjusted. 6. When eerything is working as it should, tape the newsprint to the floor and use the plumb bob to mark the location of the target screw on the newsprint. Do not alter the position of the track for the remainder of the experiment. 7. Perform the collision again, using carbon paper to mark the landing points of the spheres on the newsprint. Su07 Page 2 of 6

3 8. Put m 2 away and lower the height of the target screw so that m 1 can fly oer it with no contact as it comes off the bottom of the track. Release m 1 from the top of the track again and use carbon paper to mark its landing point on the newsprint. 9. Draw a coordinate system on the newsprint. The location of the target screw marked earlier is the origin of this system. Draw a line from here through the point where m 1 landed when it flew off the track by itself. This is the x- axis of your coordinate system (a perpendicular to this through the origin will be the y-axis). 10. Measure the distance s 1o that m 1 traeled with no collision. Record this alue and use it to calculate the elocity of m 1 before the collision. 11. Draw lines from the origin to the landing points of m 1 and m 2 after the collision, and then measure these distances s 1f and s 2f. Record the alues and calculate the elocities of m 1 and m 2 after the collision. The lines you hae drawn to the landing points are also the directions of these elocities after the collision. Measure the angle that these elocities make with your x-axis. 12. You will turn in your newsprint write the names of all group members on it. Make sure that eerything is labeled. Table 1 Collision Data m 1 (g) m 2 (g) Distance s 1o (cm) Velocity of m 1 before collision (cm/s) Distance s 1f (cm) Vertical distance d (cm) Velocity of m 1 after collision (cm/s) Free-fall time t (s) Angle between 1f and the x-axis ( ) Distance s 2f (cm) Velocity of m 2 after collision (cm/s) Angle between 2f and the x-axis ( ) Su07 Page 3 of 6

4 Analysis Since you hae masses in g and elocities in cm/s, use gcm/s as your unit of momentum. 1. Calculate total momentum in the x-direction before and after the collision (show all work). a. Before: b. After: c. Compare (% difference) the momenta from a and b. Is momentum consered in this direction? Su07 Page 4 of 6

5 2. Calculate total momentum in the y-direction before and after the collision (show all work). a. Before: b. After: c. What can you compare (% difference) to show that momentum was consered in this direction? Do so below. Su07 Page 5 of 6

6 Pre-Lab: Conseration of Momentum in Two Dimensions Name Section 1. An object falls from rest ( oy = 0) 1.00m to the floor. How much time does it take for this to happen? 2. Rather than falling straight down, the object from Question 1 is projected horizontally with some elocity o. If it traels 3.00m horizontally before striking the floor, what was the initial elocity o? 3. Consider the collision shown in Figure 1 of the handout (m 1 = m 2; 1o = 5.00m/s, 2o = 0). After the collision, m 1 has a elocity of 3.83m/s at an angle of 40 aboe the x-axis. What will be the elocity (magnitude and direction) of m 2 after the collision? Su07 Page 6 of 6