Question

• In this lab you will take data from a video and attempt to verify the Law of Conservation of Momentum. Additionally, you will take into account the uncertainty of (most of) the measurements.
  
  There is no such thing as a perfect measurement. All measurements have some amount of error. Some of that error comes from mistakes made while taking the measurement; by slightly misusing the equipment (for example, not perfectly lining up a ruler) or by misreading the equipment. As such, it is common to state the uncertainty of a measurement. This is done by using the plus/minus symbol; ±. The number following this symbol is the uncertainty. For example, the measurement "5.2 m ± 0.2 m" has an uncertainty of 0.2 m. Overall, this means: "We believe the value is 5.2 m, we acknowledge we are probably slightly incorrect, but we are supremely confident that the actual value lies between 5 m & 5.4 m."
  
  In this lab you will be asked to estimate one uncertainty yourself. The rest will either be given to you or you will calculate them using the formulas provided.
  

• Video & data table

  The video below shows a dart being fired into a cart (that is initially at rest). The beginning sequence was filmed at 240 frames per second and it took 10 frames for the dart to travel 31 cm. You can use this data to determine the dart's momentum prior to impact. For this calculation, we will assume that the values that were just stated (240 frames per second, 10 frames, & 31 cm) are all exact. Fill in the PRE-COLLISION DATA TABLE.
  
  After the collision, the frame rate is 60 frames per second. The video shows the cart (with the dart embedded in the foam block) moving forward. The frame counter has restarted so that the "X" on the cart is at 0 cm at frame zero. By advancing the video, determine at which frame the "X" reaches the 10 cm line and enter that data in the POST-COLLISION DATA TABLE. You'll also need to identify the frame uncertainty: How many frames could your data be off by? You can assume that the 10 cm & 60 frames per second are exact values.
  
  Other data that you need is stated below the video as well as the formulas for calculating speed uncertainty & momentum uncertainty. NOTE: LEAVE THE MASS IN GRAMS.
  
  Mass of dart = 6.1 g ± 0.1 g
  Mass of cart = 262.5 g ± 0.1 g
  Total mass = 268.6 g ± 0.2 g
  speed uncertainty =

  speed × frame uncertainty

  # of frames

  momentum uncertainty = mass uncertainty × speed + mass × speed uncertainty

  
  Frame uncertainty =  
  

  PRE-COLLISION DATA TABLE	# of frames needed for dart to travel 31 cm	Δt (s)	vdart,i (m/s)	vdart,i uncertainty (m/s)	pdart,i (g·m/s)	pdart,i uncertainty (g·m/s)
  Dart	10			0

  POST-COLLISION DATA TABLE	# of frames needed for cart to travel 10 cm	Δt (s)	vf (m/s)	vf uncertainty (m/s)	pf (g·m/s)	pf uncertainty (g·m/s)
  Cart + dart

• Momentum range

  Using the values you calculated for momentum & momentum uncertainty, state the range for the pre-collision momentum & the post-collision momentum in the table below.

  	minimum momentum (g·m/s)	maximum momentum (g·m/s)
  Dart (pre-collision)
  Cart + dart (post-collision)

  
  Ideally, these ranges will overlap. The claim is that momentum is conserved. Within experimental error, you have shown this is true for this collision.

• Post-collision velocity

  By applying the Law of Conservation of Momentum to this situation, derive a formula for v_f. Your answer should be symbolic (no data). Use m for the dart mass, M for the cart mass, and v for the initial velocity of the dart.
  v_f =
  
  Using the formula you just derived, determine what v_f will be for the following three cases:
  
  M = 0 (cart mass is zero); v_f =
  M = m (cart & dart have same mass); v_f =
  M >> m (the cart mass is much greater than the dart mass); v_f =
  
  Using your data and the formula you derived (at the top of this section), calculate v_f.
  v_f =  
  
  Now determine the uncertainty of this value using the following formula: v_f uncertainty =

  dart mass uncertainty × dart speed

  total mass of cart & dart

  
  v_f uncertainty =  
  
  You have now determined v_f two different ways. In the POST-COLLISION DATA TABLE you determined it using distance/time. Just above, you determined it using conservation of momentum. Using the values & uncertainties you calculated for each, fill in the table below. As before, ideally the ranges will overlap.

  	minimum vf (m/s)	maximum vf (m/s)
  Determined using distance/time
  Determined using conservation of momentum

Answer 1

Pre collision: the time is given by:

And the speed:

For the frame uncertainty we can take 5 as a reasonable value, then:

Therefore the speed with it's uncertainty is:

The momentum of the dart is:

The uncertainty:

Therefore the momentum with it's uncertainty:

After collision: The conservation of momentum equation is:

Then

If M = 0 then v_f = v, if M = m then v_f = 1/2 v and if M is much greater than m then v_f = m/M v. Using the data we get:

Using the given formula the uncertainty is:

Therefore the final velocity with it's uncertainty is:

The final momentum:

The uncertainty in the final momentum:

The momentum with it's uncertainty: