The purpose of this experiment was to prove that the impulse-momentum theorem applies to systems that involve both elastic and inelastic collisions (such as the one included in this experiment). The impulse-momentum theorem states that the change in momentum of an object is equal to the impulse applied on an object.
Apparatus:
Figure 1: Apparatus used in the experiment
The apparatus of this experiment consisted of a metal track with negligible friction, which was used to allow a cart to move in one direction on a level surface. The cart had attached to it a force senor, which was used to measure the force applied on the cart during the collisions. A motion sensor was also applied to the side of the track to measure the velocity of the cart. A computer with Logger Pro software was needed to collect and record the data from the force sensor and motion detector. To simulate a nearly perfect elastic collision, another cart with a springing piece was fixed on the other side so that upon collision with the force sensor on the moving cart, the force would be recorded and the cart would be bounced off to simulate an elastic collision. The cart was fixed onto a vertical rod that was held to the table by a C-clamp. Lastly, to simulate an inelastic collision, a wooden block with clay attached to its side was used instead of the springy cart, and this block was fixed to the same side of the track. In addition, a nail was attached to the force sensor via a rubber stopper to ensure that the cart sticks to the surface it collided with.
Procedure/Discussion (Part 1):
Once the apparatus with the springy cart was set up, the mass of the moving cart (with the force sensor attached) was measured and recorded to be 0.683 kg. Then, the force sensor was zeroed and a 500 g mass was hung on it to ensure that it is accurate in reading the actual force (for the calibration, a force of about 4.9 N). Then, the fixed cart was set up so that the hook on the force sensor directly touches the springy part during the collision. Then, a test run was made to ensure that everything is working normally, which turned out to be so.
Then, on Logger Pro, the positive direction reading on the motion detector was reversed so that the cart's initial velocity, moving away from the detector, is read as negative. Next, a specific experimental file was opened on Logger Pro so that the motion detector would record data points at a faster rate, so that the results of the experiment would turn out more accurate due to the collision being a very small amount time.
Next, the actual experiment was run. Before starting it, the force probe was once again zeroed. Then, the cart was given initial push and right after the cart started moving the data from the force sensor and the motion detector was begun to be recorded. Data was recorded until a few seconds after the collision between the two carts ended. The graphs of the velocity of the cart and the force applied by the cart are shown below:
Figure 2: Force / velocity vs time graphs of the elastic collision of the cart
As can be seen on the force vs time as well as the velocity vs. time graph, since the friction of the track is negligible and it is level, the net force exerted on the cart right before and after the collision is zero (velocity is constant). Then, the magnitude of the force on the cart is maximum when the springy piece of the fixed cart is fully compressed and is right above to go back to its natural state, which is seen as the top of the peak in the force vs time graph. To prove this, looking at Hooke's Law (F = -kx), the force of the spring is the largest when x (the distance the spring is stretched or compressed) is the largest, which is why the peak resembles max compression of the spring. Lastly, looking at the force vs time graph, the duration of the collision (the amount the peak stretches for) is about 0.18 seconds.
Then, using the force vs. time graph, the area under the peak (the time interval where the collision occurred) was found using Logger Pro. The area under that curve provides the net impulse applied on the cart. Then, using the measured mass of the cart and the velocity data obtained from the motion detector, the momentum of the cart was obtained throughout the experiment. The momentum is the mass of the cart multiplied by the velocity of the cart at each specific point in the experiment.
Figure 3: Collected and calculated data for this experiment
For example, to calculate the momentum of the cart at 0.66 s, p = (0.683 kg) (0.328 m/s) = 0.224 kg m/s. Then, the change in the momentum of the cart was compared to the net impulse applied on the cart (area under the force vs. time curve during the collision). To get the change in momentum, the momentum after the collision and before the collision had to be found. The graphs used to find them are shown below:
Figure 4: Graph of momentum & force vs time displaying impulse and momentum right after collision
Figure 5: Graph of momentum & force vs time displaying impulse and momentum right before collision
The total change in momentum of the cart during the collision can then be found by subtracting the momentum after the collision by the momentum before the collision. Doing so, the obtained change in momentum during the collision is 0.224 - (-0.264) = 0.488 kg m/s. The calculated impulse is 0.5170 N s (N s = kg m/s). As can be seen, the impulse and the change in momentum are not equal but, ruling out the error in the data due to uncertainties present throughout the experiment, are close enough (5.8% difference) to prove that the impulse-momentum theorem applies to this system and all other elastic collisions.
Conclusion (Part 1):
Looking at the velocity vs time graph shown in figure 2, the velocity is not exactly constant before and after the collision. This is due to error in the measurement of velocity cause by uncertainty in the motion detector. In addition, the fact that there is friction in the axles of the cart was never taken into account. Also, air resistance was never taken into account, which does cause the velocity of the cart to decrease slowly over time.
Also, looking at the same velocity vs time graph, the speed of the cart before the collision is not equal to the speed of the cart after the collision, showing that the collision was not completely elastic. The causes for this include the fact that the spring on the fixed cart is not ideal, therefore resulting in some energy lost to vibrations and heat in the spring. In addition, there is friction in the springy piece on the fixed cart, also resulting in loss of energy from the system, known as non-conservative work.
Looking at the force data, the force read is not zero when the cart is not colliding. This is due to the fact that the cart is experiencing air resistance, which was assumed to not be present. In addition, there is also uncertainty in the force sensor.
Lastly, it was found that there was small difference between the impulse applied on and change in momentum of the cart. This is caused by uncertainties in the measurements and the equipment used as well as the fact that the collision is not ideal and not perfectly elastic. Overall though, although the uncertainties do have some effect on the results, they are not large. They have a noticeable effect on the results because the values of the results are fairly small, so any uncertainty would have a noticeable effect on them. To lower the uncertainty in this experiment, more accurate and precise equipment would be needed as well as the use of larger measurements to result in larger results that are not as sensitive to any uncertainty.
Procedure/Discussion (Part 2):
The same procedure in part 1 was performed again for part 2, with the exception that the cart was made heavier by adding to it a 500 gram mass. The graphs displaying the velocity and force data obtained are shown below:
Figure 6: Force / velocity vs time graphs of the elastic collision of the more massive cart
As seen in the previous graphs for part 1, the net force on the cart is zero before and after the collision, and the peak represents the collision. Then, the momentum of the cart at each point was calculated and is shown below:
Figure 7: Collected and calculated data for this experiment
Then, the area under the peak of the force vs time graph was found (impulse) and the change in momentum of the cart was found using the same process (subtracting momentum after collision by momentum before collision).
Figure 8: Graph of momentum & force vs time displaying impulse and momentum right after collision
Figure 9: Graph of momentum & force vs time displaying impulse and momentum right before collision
The impulse applied on the cart was found to be 0.5820 N s and the change momentum of the cart was calculated to be 0.583 N. They are almost exactly equal (0.17 % difference), strongly proving that the impulse-momentum theorem still applies to an elastic collision using heavier objects.
Conclusion (Part 2):
As stated before, the velocity is not exactly constant before and after the collision because of error in the measurement of velocity cause by uncertainty in the motion detector, as well as the presence of friction in the axles of the cart and air resistance. Also, the speed of the cart before the collision is not equal (although very close) to the speed of the cart after the collision, showing that the collision was not completely elastic, due to the spring not being ideal and friction in the springy piece on the fixed cart, resulting in loss of energy from the system.
Again, the force read is not zero when the cart is not colliding because of air resistance and uncertainty in the force sensor. Lastly, there was a very small difference between the impulse applied on and change in momentum of the car, which is caused by uncertainties in the measurements and the equipment used as well as the fact that the collision is not ideal and not perfectly elastic. The uncertainty is very small, though, proving that this model is very accurate to simulate the impulse-momentum theorem. Stated as an improvement for part 1, the fact that higher measurement and result values was done in part 2 by increasing the mass of the cart greatly decreased the effect of uncertainty on the results.
Procedure/Discussion (Part 3):
The same procedure for part 3 was done in part 2 except the fact that the wooden block with clay attached was used instead of the cart with the springy device, and that a nail was attached to the force sensor using a rubber stopper. This was done to model an inelastic collision, where the moving object collides and sticks onto the surface it collided with, and to prove that the impulse-momentum theorem also applies to inelastic collisions. The added 500 g mass was left on the cart to give the cart a mass of 1.183 kg. The impulse of an inelastic collision would be smaller due to the fact that the collision time is smaller and that the object is stopped and not given force to move opposite its initial direction. The graphs of force vs. time as well as velocity vs time are shown below:
Figure 10: Force / velocity vs time graphs of the inelastic collision of the more massive cart
The net force is also zero before and after the collision and the peak is the collision. Because this collision was inelastic, the final velocity is zero. The momentum was also calculated and is shown below:
Figure 11: Collected and calculated data
Then, as before, the impulse was found by taking the area under the force v time graph and the change momentum during the collision was found (momentum after minus momentum before collision). The graph before the collision is displayed below:
Figure 12: Graph of momentum & force vs time displaying impulse and momentum right before collision
The reason that the graph showing momentum after the collision is not shown is simply because the momentum after the inelastic collision is zero (since the velocity is zero). The curve of the graph is different from the curve of elastic collisions since a negative force is needed to stop the moving cart. The impulse was found to be 0.3891 N s and the momentum was calculated to be 0.341 kg m/s. As expected, the impulse/momentum change of an inelastic collision is less than an elastic one. The difference between the two is somewhat large (13%) and is caused by the uncertainties and errors in the experiment, also proving that the impulse-momentum theorem applies to inelastic collisions
Conclusion (Part 3):
In this part, the velocity of the cart came out to be fairly constant before the collision (and obviously after since the velocity was zero). However, the force rad was not zero before and after the collision, which is due to uncertainty in the force sensor and the fact that the rubber stopper on the hook of the force sensor is applying some force on the hook (the mistake is that the force sensor was not zeroed after placing the nail and the rubber stopper on the force sensor). Lastly, as stated before, the uncertainty in the measurements and the equipment was the cause of the error in the difference between impulse and momentum. The uncertainty is small, but because the data and results also have small values, even the small uncertainty has some effect on the results, resulting in error. In addition, this part of the experiment was performed by only one person, making it difficult to collect data and do the experiment at the same time, resulting in larger human error. Overall, though, it can be concluded that the model for this experiment was a strong portrayal of the impulse-momentum theorem in an inelastic collision as well as an elastic collision for the previous parts. To lower the error for this part, the experiment should be done with more than one person and better equipment should be used.
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