Lab #5 -Conservation of Momentum in Two Dimensions
(revised 6/7/2014 RM)
Name: ______Date: ______
Introduction: Momentum, mass times velocity, is extremely important when analyzing the motion of a system of objects, especially when considering the results of a collision. As you learned in class, momentum is a vector, every object of a system has momentum (although it might be zero) and the momentum vectors of all of the objects in a system can be added together to give something we call the total momentum of the system. (This is similar to having several forces acting on an object and adding all these forces to get the total, or net, force acting on the object.)
Why is momentum so important? Because, as you may have learned in class, the total momentum vector of a system is constant as long as no external forces are acting on the system or as long as the total external force acting on the system is zero. This is the law of Conservation of Momentum. This does not mean that each object in the system will have a constant momentum, but that the total of all momenta added together will be constant. Individual momenta may change, but the total will remain the same as time goes on.
A system might actually consist of a single object. If there are no forces acting on the object then the object’s momentum will be constant. One source of confusion could be that even though the object might be moving, its momentum is still constant. Remember that momentum is mass times velocity so if velocity is constant so too is momentum. One way to see this is to draw a dot to represent the location of the object at various points in time and at each dot draw an arrow to represent the object’s velocity, or momentum, with the tail of the vector on the dot (see figure 1). Even though you have drawn several dots with an arrow on each, all these arrows (vectors) are equal because their lengths are all equal and also their directions are equal. So you do not really need to draw an object’s momentum vector with its tail at the location of the object. All you really need is a single vector drawn somewhere on your paper that represents the direction and magnitude of the momentum at all time. Only when the momentum changes should you draw separate momentum vectors at different locations on your paper.
Figure 1
In this lab you will analyze several collisions that take place in two dimensions. The momentum vectors will therefore have up to two components. In all of these collisions one ball will be moving and the other will be stationary before the collision. How do we determine the velocity, or momentum, of the moving ball before the collision? We will do this by letting the ball roll down a ramp as shown in Figure 2. When the ball gets to the bottom of the ramp, just before colliding with the stationary ball, it will have a speed that can easily be determined as outlined in Part I.
Figure 2
Part I: Initial Momentum of Moving Ball and Initial Momentum of the System.
In this section blue is graphing activity, and green is data activity.
Clamp your ramp to the edge of a table so that the end of the ramp with the adjustable ball holder is hanging off the edge of the table. Tape a large piece of paper (approximately 3 ft by 3 ft) onto the floor so that one edge of the paper is directly below the edge of the table.
Drop a plumb line down from the end of the ramp, centered on the ramp,so that the plumb bob touches the paper. Mark this spot on your paper and label it with thenumber 1.
This point will help define the initial direction of travel of the moving ball and it will indicate the horizontal position of the moving ball just before the collision.Your instructor will demonstrate this. You will use carbon paper to determine where the balls land after the collision.
You might ask why we are letting the balls fall when we are really trying to study a two dimensional collision? Letting the balls fall introduces a third dimension. Shouldn’t we just let the balls roll on a flat horizontal surface so that the motion is entirely in the horizontal plane? We could do this if we had a perfectly horizontal surface and rolling friction was negligible, but we would still need to figure out a way to measure the balls’ speeds. Our solution is to use the acceleration due to gravity along with the vertical and horizontal distances traveled to determine the balls’ velocities immediately after the collision. This is just like the projectile motion lab where we fired a ball horizontally to determine its launch speed. We will thus minimize any error due to a horizontal surface not being perfectly horizontal,eliminate any error due to rolling friction, and reduce any difficulty in measuring horizontal speeds directly.
The velocity of the moving ball before the collision multiplied by the ball’s mass will give the ball’s momentum before the collision. Since the other ball is stationary the moving ball’s momentum is equal to the system’s total momentum and this total momentum will equal the sum of both balls’ momenta just after the collision.
To measure the moving ball’s initial speed we will let the ball roll off the ramp without colliding with the other ball. As soon as the ball leaves the ramp it will start falling under the force of gravity. The distance from the ball’s landing spot to the spot directly below the ramp that you marked previously will equal the ball’s horizontal distance traveled during free fall.
Measure the mass of each steel ball. They should be the same, and we will assume they are the same for the purpose of this lab. Enter these masses in the data section (m).
Allow a ballto roll down the ramp several times, starting from the same initial height each time, and use taped down carbon paper to mark the ball's landing spots. Determine a point on the paper that represents the average of these landing spots and label this point with the number 2.
Draw a line from this average landing spot, point 2, to the spot under the edge of the ramp, point 1. This line represents the direction of the moving ball’s momentum or velocity. Measure the length of this line and record it in the data section (X).See figure 5. This line will also be our x-axis. Our y-axis will then be perpendicular to this line, with the origin at point 1.
To determine the ball’s horizontal speed we need to know how much time it took for the ball to travel this distance. This time is equal to the time it took for the ball to fall the distance it fell. Measure the vertical fall distance and record it in the data section (H). (From where do you measure the initial height of the ball?)
Now calculate the time it took for the ball to fall and enter this in the data section (T).
Nowcalculate the horizontal speed of the ball and enter it in the data section (VX). Be sure to show your calculations and do not round any calculation until you have calculated the horizontal speed. (Keep at least four decimal places in all your calculations and remember to include as many significant figures as possible in all of your measurements.)
Finally, calculate the ball’s horizontal momentum (Px , or Ptot), andexpress it in unit vector notation in the data section. Remember that this momentum is the total momentum of the system before and after the collision that we will now consider.
Part II: Collision With Identical Ball at about 20 degrees.
Now adjust the stationary ball holder so that the moving ball collides with the stationary ball at a slight angle of about 20 degrees as shown in figure 3. This angle is not critical, we just don’t want a head-on collision, nor do we want a glancing collision.
Figure 3 / Figure 4Also, adjust the height of the stationary ball holder so the balls are at the same vertical position when they collide. Be sure to adjust the stationary ball holder so that the center of the moving ball is right at the edge of the ramp when it just touches the stationary ball.
Doing this will ensure that the moving ball is directly over point 1, on your paper, when the collision occurs. Tighten the ball holder in place. See Figure 3 and 4.
Drop a plumb line directly below the stationary ball holder to make a mark on the paper directly below the initial position of the stationary ball. Label this point “S1” (S for stationary).
Now place the stationary ball on the ball holder and hold the moving ball at the top of the ramp (where you let it go before). Do a trial collision to see where the balls will land. Tape down pieces of carbon paper at the landing locations and do three trials to create three landing marks for each ball. Find an average for these locations and label these points “S2”(for the initially stationary ball), and “M2”(for the initially moving ball).
Draw a line between S1 and S2 (call this S), and between point 1 and M2(call this M)as in figure 5. These lines give the two balls' distances traveled horizontally after the collision.
Measure the lengths of these lines (S and M) and enter them in the data section.
Figure 5
To determine the speeds of the balls just after the collision we simply divide the above distances by the time of fall for each ball. Does each ball take the same time to fall? Are these times the same as the time calculated in Part I? Calculate the speeds of each ball immediately after the collision and enter these speeds in the data section. Calculate the momenta of the balls just after the collision and enter these momenta in the data section.
Part III: Analysis of Data From Part II.
To verify that momentum is indeed conserved we can use either an algebraic method, or a graphical method. We will do both!
Algebraic Method: This method involves breaking vectors into their orthogonal components and showing that the vectors' x-components and y-components are separately conserved.
Put another way, the initial momentum of the system had only an x-component (PX, which you calculated in Part I). The x-components of the two balls after the collision should add up to equal PX, and the y-components should add up to equal zero (i.e. the y-components should be equal in magnitude, but opposite in direction).
*****
The lines S and M were measured as distances, but these lines can also be equivalently thought of as the momentum vectors of the two balls just after the collision.
This is true because we determined momentum by dividing these distances by drop times and then multiplying them by their masses. Since the drop time and mass are equal for both balls, the two balls' momenta are different only because the distances they travelled are different.
So the horizontal distances travelled by the balls give us an equivalent measure of their momenta before and after the collision.
Read the last three sentences again, as needed, to be sure that you understand what is being explained.
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Measure the angle of line S relative to the x-axis (θ). Notice that points S1 and 1 are not coincident.
See figure 5. Enter this angle in the data section.
Measure the angle of line 1-M2 relative to the x-axis (φ). See figure 5. Enter this angle in the data section.
Calculate the x component of the stationary ball (S cosθ). Enter this result in the data section.
Calculate the y component of the stationary ball (Ssinθ).Enter this result in the data section.
Now calculate the x component of the moving ball (M cosφ). Enter this result in the data section.
Calculate the y component of the moving ball (M sinφ). Enter this result in the data section.
If momentum is conserved in the x direction, X = S cosθ + M cosφ. Is this true? State your answer in the data section.
If momentum is conserved in the y direction, S sinθ = M sinφ. Is this true? State your answer in the data section.
Graphical Method: This method involves using actual distance measurements to see if the x and y momenta of the 2 ball system is the same before and after the collision (linear mometum is conserved).
On your graph paper, measure the distances X1, X2, Y1 and Y2, as depicted in figure 5. Enter these distances in the data section.
Also, enter the measurement X from Part I in the data section.
Is X = X1 + X2? State your answer in the data section.
Is Y1 = Y2? State your answer in the data section.
State two non-trivial systematic errors for this experiment in the data section.
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Data
Part I: No Collision
Mass of the first ball (m) = ______kg
Mass of the second ball (m) = ______kg
Horizontal Distance Traveled of Moving Ball (X) = ______m
Vertical Distance of Fall (H) = ______m
Time to Fall (T) = ______sec
Horizontal speed of Moving Ball (V) = ______m/sec
Ball’s Horizontal Momentum (Px , or Ptot ) = ______kg-m/sec
Show calculations here:
Part II: Collision With Identical Ball at about 20 degrees.
Algebraic Method:
Distance from S1 to S2 (S) = ______m
Distance from point 1 to M2 (M) = ______m
Angle of line S relative to line X(θ) =______ degrees
Angle of line M relative to line X (φ) = ______degrees
What is x component of the stationary ball (S cosθ) = ______m
What is y component of the stationary ball (S sinθ) = ______m
What is x component of the moving ball (M cosφ) = ______m
What is y component of the moving ball (M sinφ) = ______m
Show calculations here:
Conclusions:
IsX = S cosθ + M cosφ?
If not, why not?
Is S sinθ = M sinφ?
If not, why not?
Graphical Method:
X1 = ______m
X2 = ______m
Y1 = ______m
Y2 = ______m
X = ______m
Show calculations here:
Conclusions:
Is X = X1 + X2?
If not, why not?
Is Y1 = Y2?
If not, why not?
State two non-trivial systematic errors in this experiment.
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