Impulse and Momentum

What’s the Big Idea‽

This lab explores the Impulse-Momentum Theorem and the relationship between impulse and momentum.

Procedure

On the technical side of things, we needed a Vernier LabQuest, a laptop, a motion detector, and a force probe. On the less technical side of things, we needed a dynamics cart, a frictionless track, a 500 g mass, some string, a rubber band, and a rod with its accompanying stand.

To set up, we placed the cart on the track, with the motion detector on one side and the force probe secured on the rod on the other side. The end of the cart was attached to a rubber band which was attached to the string which was attached to the force probe.

The basic setup. 

Another view of the setup.

After setting everything up, we measured the mass of the cart (which was 0.51 kg) and opened Logger Pro. We calibrated the force probe and began collecting data, pulling the cart away from the force probe and letting it bounce back. The combined use of the force probe and the motion detector allowed us to measure position, time, velocity, and force, as shown in this graph:

The measured data.

What’s fantastic about Logger Pro is that it allows us to easily calculate and visualize the impulse given to the cart by taking the integral of the curve.

Takeaways

From this lab, we learned that impulse is not only equal to the change in momentum, but the integral of force over time. Additionally, specifically regarding this experiment, a higher spring constant in the rubber band yields a greater change in momentum and therefore a greater impulse.

Momentum and Collisions

What’s the Big Idea‽

The following lab focuses on…you guessed it, momentum! And to that end, we’ll also be playing with the momentum of a system before and after internal collisions. Stay tuned and we’ll perform experiments that not only answer the questions “What is momentum?” and “How is it calculated?”, but also demonstrate conservation of momentum.

Procedure

Like many of our labs, this lab revolved around the low-friction dynamics cats.

That’s dynamic carts, not cats.

We set up two dynamics carts on each side of the track, behind each of which was a motion detector. As usually, a Vernier LabQuest recorded the data onto a laptop. The following photos demonstrate this described setup:

Say hi to Jill!

The motion detectors recorded how the carts’ velocity changed over time.

In addition to using the motion detectors to record the carts’ velocity, we used a scale to measure the carts’ mass.

Cart 1 Pushed; Cart 2 Not Pushed
	Cart 1			Cart 2			Total
	mass	velocity	momentum	mass	velocity	momentum	mass	velocity	momentum
before	0.52 kg	0.13 m/s	0.07 N/s	0.51 kg	-0.00 m/s	-0.00 N/s	1.03 kg	–	0.07 N/s
during		0.12 m/s	0.06 N/s		-0.00 m/s	-0.00 N/s		–	0.06 N/s
after		-0.00 m/s	-0.00 N/s		0.10 m/s	0.05 N/s		–	0.05 N/s

If we divide the difference of the final momentum and the initial momentum by the average momentum, we can find the percent difference. The first trial yields a percent difference of 15.4%, which while not fantastic, does suggest the conservation on momentum. The higher-than-expected percent difference is most likely due to precision errors incurred when rounding the measured values.

Cart 1 and Cart 2 Pushed at Same Velocity
	Cart 1			Cart 2			Total
	mass	velocity	momentum	mass	velocity	momentum	mass	velocity	momentum
before	0.52 kg	0.12 m/s	0.06 N/s	0.51 kg	-0.13 m/s	-0.07 N/s	1.03 kg	–	-0.01 N/s
during		0.05 m/s	0.03 N/s		-0.01 m/s	-0.01 N/s		–	0.02 N/s
after		-0.09 m/s	-0.05 N/s		0.12 m/s	0.06 N/s		–	0.01 N/s

Similarly to the last trial, we can find the percent difference. However, this trial yields bizarre results because the average momentum is zero, making it impossible to find the percent difference.

Cart 1 and Cart 2 Pushed at Different Velocity
	Cart 1			Cart 2			Total
	mass	velocity	momentum	mass	velocity	momentum	mass	velocity	momentum
before	0.52 kg	0.40m/s	0.21 N/s	0.51 kg	-0.30 m/s	-0.15 N/s	1.03 kg	–	0.06 N/s
during		0.39 m/s	0.20 N/s		-0.29 m/s	-0.15 N/s		–	0.05 N/s
after		-0.24 m/s	-0.12 N/s		0.33 m/s	0.17 N/s		–	0.05 N/s

The percent difference of the third trial is 18.2%, which is worse than expected. Again, I believe this is due to precision errors.

Takeaways

Although the data isn’t fantastic, it roughly shows us how momentum is conserved with at least some degree of certainty. Before the collision, the total momentum of the system can be calculated as the sum of the carts’ individual momenta (as momentum is a vector quantity). And in the same manner, the total momentum of the system can be calculated to determine the difference between the momenta before and after the collision.

Spring Constant Lab Challenge: SELF DESTRUCT INITIATED

Secure Briefing

Is it just me or does this make it look like physics is espionage?



Especially that last line. The drama.

What’s the Big Idea‽

Excuse me, what’s the big idea‽ How dare you ask such an insulting question on such an important mission. The big idea is that we have a limited time to determine what the spring constant of the dynamic cart’s plunger! There’s no time to waste!

Procedure

First question: what tools do we need to determine the spring constant we’re looking for? Well, we can use the formula F = -kx to do so. Therefore, we need to apply a force onto the cart and measure that force, measure the total displacement that it goes through.

To do so, we used the Vernier LabQuest, a force sensor (with a rubber attachment), a dynamics cart, a meterstick, and a calculator, set up as below:

The calculator’s role was especially important, albeit unorthodox.

Next, we unlocked the plunger and pushed it against the force sensor to measure the amount of force applied, allowing it to go just far enough that the plunger doesn’t lock. A displacement of -0.028 m was measured using the meterstick, with the other data as follows:

The maximum force, as indicated by this graph, is 18.14 N, which corresponds to the maximum compression we obtained above. Next is simple substitution; just use the equation F = -kx and solve for k, then bam! You’ve got a value of 648 N/m for the cart’s spring constant.

Takeaways

A mission well done. We now know that our physics prowess is competent enough to calculate the spring constant of the dynamic cart’s plunger!

Energy of a Tossed Ball

What’s the Big Idea‽

When is energy conserved? When is energy not conserved? And how can we measure that energy? This lab explores these questions by investigating what happens to the energy of a tossed ball.

Procedure

First, we gathered the materials for the lab, which included a Vernier LabQuest and laptop, as usual, but also included a motion detector and a ball (for tossing purposes).

The completed setup.

Once everything was connected, we placed the motion detector flat on the table, tossed the ball directly upwards, and recorded the corresponding data.

The raw data from the LabQuest.

A much clearer readout of the above data.

Oh, and we also measured the ball’s mass with a scale:

mass

0.403 kg

The following table is a synthesis of three recorded points (one corresponding to after the release, one at the top of the path, and one before the catch), yielding data for the time, height, and velocity. This then allowed us to calculate the potential energy (using the formula PE = mgh), kinetic energy (KE  = ½mv²), and mechanical energy (ME = PE + KE).

position	time	height	velocity	PE	KE	ME
after release	1.299 s	0.375 m	2.580 m/s	1.482 J	1.341 J	2.823 J
top of path	1.598 s	0.805 m	0.002 m/s	3.182 J	0.000 J	3.182 J
before catch	1.865 s	0.464 m	-2.580 m/s	1.834 J	1.341 J	3.175 J

If the tossed ball demonstrates conservation of energy, only conservative forces (e.g. gravity) are acting on the system. Otherwise, a combination of conservative (again, e.g. gravity) and nonconservative forces (e.g. friction) are acting on the system. The calculated change in mechanical energy suggests conservation of energy, however, I think that the recorded point directly after release is incorrect. The change from 2.823 J to 3.182 J would only occur if an upward force was temporarily acting on the ball (such as a hand), therefore, this point is unsupportive of our conclusions. (However, in a way, it does show that the hand pushing up on the ball is somewhat like a nonconservative force, because it only acts in one direction.)

Ignoring the first data point, the change in mechanical energy from 3.182 J to 3.175 J makes much more sense. For the most part, the total energy of the system remains constant, suggesting that the most significant forces acting on the system are conservative. I presume the slight decline in energy can be attributed to friction in the form of air resistance (a nonconservative force) acting on the ball.

Takeaways

In this lab, we learned about conservative and nonconservative forces. Basically, as I just described, we can determine whether or not a system demonstrates conservation of energy by measuring its change in energy over a period of time during which forces are acting on it. If the change in energy is minimal, work that displaces an object is done by conservative forces. Contrastingly, if the change in energy is significant, this displacement is partially being done by nonconservative forces.