Lab 14(13th): Impulse-Momentum

Title: Impulse-Momentum activity

Purpose: Verify that the impulse-momentum relationship holds true in a variety of situations.

Theory:
Impulse and momentum are directly related to each other where Ft=mv for an object such that some value of impulse will produce the same numerical change in momentum on an object. This relationship can be derived using kinematics equations where a given force will accelerate an object of mass m at a rate of F/m, the total change in velocity over a time interval t will be equal to (F*t)/m, and then the change in momentum is m((F*t)/m) which equals F*t, or impulse. By independent measurements of force, time, mass, and velocity of an object, impulse and momentum may be calculated and compared to verify their relationship in elastic and inelastic collisions.

Apparatus:

• Spring loaded cart
• Cart with force sensor attachment
• Cart track
• C-clamp
• Clamp
• Metal rod 
• Motion detector

Procedure:

Level the cart track horizontally with one end near the edge of a table. Measure the mass of the cart with the force sensor attached. Using the C-clamp, metal rod, and clamp, secure the spring loaded cart at one end of the cart track so that the force sensor on the other cart will collide with the extended spring. Position the motion detector behind the force sensor cart. Lightly push the cart and collect data for position and force vs time. With the same setup, add more mass to the cart and collect the appropriate data. Then, replace the spring loaded cart with a clay mass and attach a sharpened screw to the force sensor. Push the cart so that it collides and sticks into the clay while collecting the appropriate data.

Data and Graphs:

Photo of the experimental setup

Mass of cart, initial and final velocities used to calculate change in momentum

Mcart	0,648	kg
Mcart2	1,148	kg
Velocities (m/s)
Experiment 1
Vi	Vf
0,421	-0,296
Experiment 2
Vi	Vf
0,404	-0,262
Experiment 3
Vi	Vf
0,619	0

Graph of Force vs time and Velocity vs time for first collision, integral of force taken to find impulse

Graph of Force vs time and Velocity vs time for faster elastic collision

Note: graphs not aligned by Time axis

The clay target used to simulate an inelastic collision

Graph of Force vs time and Velocity vs time for inelastic collision with clay

Comparison of calculated momentum and force integrals from graphs

Δp=m Δv %error = (true value - experimental value)/true value *100 Calculated momentum (kgm/s) Force integral (Ns) % Error Experiment 1 -0,465 -0,4866 -4,52 Experiment 2 -0,765 -0,8252 -7,35 Experiment 3 -0,401 -0,4070 -1,45

Data Analysis:

The initial and final velocities of the carts for each experiment were found by locating the points at which a force was exerted and the points at which the force stopped being exerted. From these values, the change in momentum was calculated according to the equation Δp=m Δ. Percentage error values were calculated using the force integral given by LoggerPro as the "true" value.

Conclusion:

For the first and third experiments, the percentage error between the force integral and the calculated momentum were both under 5% while the second experiment had a larger error at -7.35%. Overall, the calculated moments were all slightly lower values compared to impulse. Because the duration of each collision was very short at approximately 0.2 seconds for elastic collisions and 0.1 seconds for the inelastic collision, friction does not have a significant distance to act over and is thus negligible. The lower values of change in momentum may then be due to inaccuracies in finding the initial and final velocities. The force sensor was capable of producing a larger number of samples per second than the motion detector. This means that the force graphs will produce more accurate results as the entire collision will be more accurately described. In terms of finding the velocity of the object, this means that a corresponding point on the force graph may not have an exact time equivalent on the velocity graph, leading to uncertainties in calculated momentum. Assuming uncertainties in velocity of +-0.01m/s and an uncertainty in mass of 1g, the propagated uncertainties are shown below.

(dp/p) = (dm/m) * ((dv1/v1) + (dv2/v2))
dp = p ((dm/m) * ((dv1/v1) + (dv2/v2)))
Uncertainty in Momentum
Experiment 1
0,00719
Experiment 2
0,00893
Experiment 3
0,0100

The only value of impulse to fall within the uncertainty range of momentum is that of experiment 3, suggesting that is it probable that at least one of our idealized assumptions is false, that our method of data collection is inadequate, or that our experimental setup is flawed.