Lab 7 Collisions - Elastic & Inertia CPI F23(1) (1)

7 Collisions and Interia Purpose Students will create collisions in order to observe how collisions a ↵ ect the motion of objects, to experimentally determine the inertia of an object, and to explore the conservation of linear momentum as a fundamental law of nature. Equipment • Metal or Plastic Track - PASCO 1m with End Stops • Smart Carts (1-Red and 1-Blue) with Magnetic Bumpers • Computer with CAPSTONE Software 7.1 Inertia A collision is an interaction between two objects that occurs during a very short time interval - so short that the interaction with anything outside of the system is negligible compared to the interaction between the two objects. In this activity we will investigate the e ↵ ects of collisions by observing the motion of two carts before and after a collision. An object’s motion is highly dependent upon its inertia: its tendency to resist any change in velocity ( m/s ). We will observe experimentally that during a collision, the ratio of the inertias ( m ) of the colliding objects equals the negative of the inverse ratio of their changes in velocities ( m/s ). m 1 m 2 = - Δ v 2 Δ v 1 (17) 7.2 Collecting Data 1. Log into Blackboard and locate the lab 7 folder and find the ”Collisions and Inertia” CAPSTONE file. 2. Download and open the file. 3. You will need to record the velocities for cart 1 and cart 2 in your lab manual. The velocity for each cart is displayed as a velocity time plot. Remember to record positive and negative values based on direction of motion of the carts. The lab equipment is set up so carts moving to the right have a positive velocity and carts moving to the left have a negative velocity. 4. You will need to turn on both carts and connect them to the software. If you do not remember how to do this, refer the handout in the lab folder 7, ”Connecting a SMART Cart to CAPSTONE.” 5. To record data, click ”Start” and give the carts the desired push to achieve the desired collision. 67

6. Click ”Stop” to stop after the desired collision occurs. 7. Click the ”Scale axes to show all data” button to scale the plot to the window. 8. Next, use the ”Add Coordinates” tool and select ”Add Multi-Coordinates Tool” to add a slider bar at the bottom of the plot. You should notice as you move the slider back and forth, using the mouse or the arrow keys, that you are highlighting one data point for the red cart and one corresponding data point for the blue cart. 9. Select a point just before and after the collision, record the velocities for both carts at both positions. 68

C: From Behind Collisions - Give the Red cart a small push so that it moves to the right, Give the Blue cart a larger push so that is catches up to and collides with the Red cart Diagram of Collisions Before 1 After This collisions may take some practice. Give one of the carts a small push quickly followed by giving the other cart a bigger push in the same direction. Make sure they collide and interact before one of the carts hit the track bumper. Record the initial and final velocities for both Carts in Data Table 1. Data Table 1 Trial Blue v 1 i ( m s ) Blue v 1 f ( m s ) Red v 2 i ( m s ) Red v 2 f ( m s ) Stationary Head-On From Behind 7.2.2 Data Analysis Calculate Δ v 1 and Δ v 2 and the ratios Δ v 2 Δ v 1 and m 1 m 2 record your values in Data Calculations Table 1. Data Calculations Table 1 Trial Blue Δ v 1 = v 1 f - v 1 i Red Δ v 2 = v 2 f - v 2 i - Δ v 2 Δ v 1 m 1 m 2 Stationary Head-On From Behind 70

According to Equation 17, the ratios of - Δ v 2 Δ v 1 and m 1 m 2 should be equal in magnitude. 1. Which type of collision gave the least di ↵ erence between the two ratios? Why 2. Which type of collision gave the greatest di ↵ erence in the two rations? Why? 7.3 Determining an Unknown Inertia Experimentally Using our observations about the relationship between inertia and changes in velocity during a collision, we can determine the value of an unknown inertia experimentally by colliding it with a known inertia. Let m 1 be a known inertia and m 2 be unknown. Solve Equation 17 for m 2 in terms of m 1 and the changes in velocity. Add the unknown mass found at your station to the top of the Red Cart so that it’s inertia is increased by an unknown amount. This will replace m 2 as m 2 + m unknown in your equation. Pick one of the above collision types and repeat it with the Blue and Red Carts. Record your data in Data Table 2. Data Table 1 Trial Blue v 1 i ( m s ) Blue v 1 f ( m s ) Red v 2 i ( m s ) Red v 2 f ( m s ) 71

2. Expand the Δ v 0 s in terms of final and initial velocities for each cart. 3. Multiply the expanded terms. 4. Finally, rearrange the equation so that initial terms are on the left, final terms are on the right. This final equation is called the law of conservation of linear momentum. Classically, physicists define the product of inertia ( m ) and velocity ( v )as an object’s linear momentum: -! p = m -! v (18) With this definition, your equation in step 4 becomes: -! p 1 i + -! p 2 i = -! p 1 f + -! p 2 f (19) The total momentum of all carts involved in the collision is therefore conserved: the total momentum after the collision equals the total momentum before the collision, (at least when all other interactions, such as friction, can be ignored). Let’s test this idea of con- servation by using your velocity data from before to calculate the momentum of each cart before and after each collision and record the values in Data Calculations Table 3. Let the Blue SMART Cart be Cart 1 and the Red SMART Cart be Cart 2. Make sure that you use the correct sign for the velocity, momentum is a vector quantity. Now calculate the total initial momentum, the total final momentum, and the di ↵ erence between these total momenta for each collision to see if the values are the same. 73

Data Calculations Table 3 -! p 1 i -! p 1 f -! p i = -! p 1 i + -! p 2 i -! p 2 i -! p 2 f -! p f = -! p 1 f + -! p 2 f | -! p f - -! p i | Stationary Collision Head-On Collision From Behind Collision Unknown Inertia Collision 7.4.1 Results Discussion & Error Analysis 1. According to your results in this laboratory exercise, is linear momentum conserved? Provide proof from your data calculations. 2. What are sources of error in this experiment? Name the error type, what caused it, and how it might have a ↵ ected your results. Lab Instructor Signature Score / 10 points 74