Lab 3 Worksheet - Equations of Motion (Student)

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San Diego State University *

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195L

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Physics

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Apr 3, 2024

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Lab 3: Equations of Motion San Diego State University - Department of Physics Physics 195L – Fall 2022 Theory Most people are familiar with the idea of objects in motion. We drive cars, play catch and watch objects fall. But it was not until the sixteenth and seventeenth centuries that our modern understanding of motion was established with contributions from Galileo Galilei and Isaac Newton. In this section we will review the physics that describes objects in motion. All kinematic equations can be derived from the basic equation of kinematics, , which says that when a force is acting on an object with mass , that object experiences an acceleration . To see how this equation yields other equations of motion, we need to first understand the relationships between acceleration, velocity , and displacement . Displacement is the easiest of these quantities to measure, since we can measure it with a meter stick; it is just the change in position of an object. If an object’s position is changing with time, then we say it has a nonzero velocity, and that velocity is, on average, given by Similarly, if an object’s velocity at any given moment is changing with time, then we say it has a nonzero acceleration, and that acceleration is, on average, given by These equations are useful for averages, but they fail to answer the question: what is the exact acceleration, velocity, and position of the object at the time ? To answer this question, we need to describe acceleration, velocity and position not as averages, but as functions of time: These functions are related to each other through the limit in the averages given above. You may know that such limits are called derivatives; if you have not taken calculus this is not required information however. Speed is nothing but the derivative, or algebraically the instantaneous rate of change, of position: 1

PHYSICS 195L LAB REPORT - LAB 3: EQUATIONS OF MOTION Acceleration is the derivative of speed (the instantaneous rate of change of speed): Constant acceleration Knowing how , , and are related, we can consider a specific physical example. One of the most common examples explored in this course is a system with constant force. A quantity which does not change with time is called a constant. In other words, for whatever variable we are claiming is constant. Recall that , so if is a constant, then acceleration must also be constant: What happens to the speed and position functions when acceleration is constant? To find out, let’s first name the constant acceleration for brevity. Using the equation from above, we know that acceleration is the derivative of speed: We can invert this equation by integrating from 0 to : which yields , or . We can repeat the same process for the relationship between speed and position: Integration yields: , or Procedure In this lab we will check whether or not the equations of motion we derived in the theory section actually work. To test these equations, we will conduct an experiment where a cart rolls down a track with constant acceleration. By recording the speed and position of the cart as a function of time, we are able to visualize the functions and . Then, using curve fitting, we will be able to test whether or not the functions we observe match those predicted by the theory. 2

PHYSICS 195L LAB REPORT - LAB 3: EQUATIONS OF MOTION Set up 1. Connect the Smart Cart to PASCO™ via Bluetooth in the Hardware Setup Window (located to the left). 2. Set up the track as shown in Figure 1 with an end-stop located at both ends of the track. 3. Attach the Smart Cart’s magnetic bumper (if you do not have a magnetic bumper, press the trigger on top of the cart to extend the plunger). 4. Use two Stackable Masses to incline the track. Place the Smart Cart on the track and make sure that the magnetic bumper or plunger faces downhill. 5. Position the back of the Smart Cart at the top of the track. This is the starting position for the Smart Cart. Figure 1: The Smart Cart has the plunger extended on an elevated track by two masses. Part 1: Position vs. Time 1. Click on Record and then release the cart. Note that data collection will not start until the cart travels at least 10 cm due to the initial starting condition. This will give you better looking data. 2. You can stop recording at any time, but there is an automatic stopping condition that stops the recording after the cart travels 80 cm. You can delete unwanted runs using the Delete Run feature in the Experiment Control Bar. 3. Use the highlighter tool to select only data recorded while the cart is in motion up until it reaches the end of the track. 4. Select the User Defined Curve Fit from the Graph Tool Palette. 5. The curve fit will currently look like . You can edit this (in the Curve Fit Editor on the left) and change it to . You can either type ‘o’ for and or you can right click and add a subscript. Capstone can read both. 6. Click on Apply. 7. Record the initial position , the initial velocity , and the acceleration in Table 1.1. 8. Include your Position versus Time in Figure 1.1 in the Data section. Right click on the edge of the PASCO™ graph object and select “Copy Display”. Paste into this document with “Ctrl+v”. Part 2: Velocity vs. Time From the same run, we can also look at the velocity as a function of time. 1. Select a User Defined Curve Fit from the Graph Tool Palette. 2. The Curve Fit will currently look like . You can edit this (in the Curve Fit Editor on the left) and change it to 3. Click on Apply. 3

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