Motion in One Dimension

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Jerome Blaise Gaines
5 years ago
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1 Motion in One Dimension Motion in one dimension is the simplest type of motion. Distance, velocity and acceleration are all vector quantities and must be treated as vectors. However, in onedimensional motion the sign of the quantity can indicate the vector nature of the quantities. One direction is assumed to be positive then the opposite direction is consequently negative. If we assume the acceleration is constant for a time interval, then the velocity is changing at a constant rate. This implies that the curve of a graph of the velocity vs. time is a straight line. The graph below (Fig. 1) is set up that at t = 0, v = v o (initial velocity) and at a latter time t the velocity is v. The area under the curve is equal to change in position of the object. v a v t (v v o) t v v o at (1) v o x (x x o ) area v o t 1 2 (v v o)t Fig. 1 t x x o v o t 1 2 at2 (2) A plot of distance vs. time gives a parabola (a polynomial of degree 2). The slope of the line is the velocity. x Eqns 1 and 2 are called the equations of motion. They give the position and velocity of an object moving with constant acceleration as a function of time. x o Fig. 2 y If Eqn. 1 is solved for time and substituted into Eqn. 2, then time is eliminated and results in Eqn. 3 v 2 v o 2 2a(x x o ) (3) In this lab, the motion of a cart on a track will be used to demonstrate the usefulness of the above equations. Apparatus: Cart and track Ultrasonic Transducer 8

2 Lab Pro Procedure: Case 1- Cart on a horizontal track Blocks to elevate track 1. Connect the Ultrasonic transducer to the lab Pro (Dig/Sonic 1) connection. Line up the transducer so the speaker/reciever points down the track. Be sure the switch on the transducer is set to cart. Open the program Lab Pro/Experiment/Physic with computer/carts. The cart needs to be 20 cm from the speaker/reciever. The ultrasonic transducer works by having a metal plate in the speaker vibrate and emitt ultrasonic pulses. The plate then becomes the reciever and starts to vibrate again when the reflected sound hits it. By measuring the time interval between when the pulse leaves and then returns and knowing the velocity of sound, the distance between the ultrasonic transducer and the cart can be calculated. However, the reciever picks up any relected sound including reflections from your hand or the track. Ultrasonic Transduce Fig Start collecting data by clicking on the collect button at the top of the screen. With your finger or a pencil, give the cart a small push. Click the collect button on the screen a second time to stop collecting data. On the screen, the position and velocity of the cart versus time should be displayed. There are at least 3 regions on the graph. Region 1 is where the cart is at rest (v =0) and x = x o, region 2 is where you give the cart a push (v is increasing and x is increasing) and region 3 is where the cart is coasting (v is decreasing and x is is increasing). There may be a region 4 where the cart stops or hits the end of the track. 3. The Logger Pro can also be used to fit a curve to the graph. On the position versus time graph, hi-lite region 3. (That is the region where the cart is coasting.) At the top of the menu, click on Analyze/curve fit or click the f(x) = button. The curve fit menu will appear. Choose the button for a quadratic equation and click try fit. A black line should appear and you will be able to see how good the curve fits the data. If it is a good fit, then click ok. A box on the graph will appear that gives you the values a A, B, and C for the curve fitting equation x = At 2 + Bt + C. 4. On the velocity versus time graph, hi-lite region 3, but this time fit the curve to a linear equation v = mt + b. 5. From the curve data, find the initial velocity and acceleration for region 3 by using the curve fit data from position versus time graph and then from the velocity versus time graph. Record the values in the data table. 6. On the position versus time graph, choose from the menu Analyze/examine or click on the x= button. This function will generate a box that shows the data points for x and t 9

3 for any position. Using this function, find the position of the cart where it first starts to move and also the position where the cart stopped. The difference gives you the distance the cart moved. Now on the velocity versus time graph, hi-lite the region from where the cart first starts to move until it stops. Click on Analyze/integral or button with a curve and two vertical lines. This function calculates the area under the curve which for a velocity versus time graph is the distance the cart has traveled. Record the values in the data table. 7. Print out the graphs showing the boxes for the curve fits and the integral. Case 2 Cart moving down an incline plane Ultrasonic Transduce Fig. 4 x y 8. Place a block under the end to the track that is opposite the motion detector. The head of the motion detector can be tilted so that it points up the incline. Click collect and then release the cart at the top of the incline. Catch the cart at the bottom of the incline. On the graphs, identify the reqion from where you released the cart to where you caught the cart. Repeat all the curve fitings and integral that were done before and record the values in the data table Case 3 Cart moving up an incline plane 9. Place the cart on the bottom of the incline. Click collect and give the cart a push up the incline so that the cart travels up the incline, stops and then returns to starting point. Catch the cart at the bottom of the incline. Repeat all the curve fittings and integral that were done before and record the values in the data table. Case 4 Determining the acceleration of gravity 10. Measure the distances x and y as shown in Fig. 4. Using = tan -1 (y/x) and a=g sin calculate values for g from the accelerations measured in case 2 and 3. 10

4 Data: Case 1- Cart on a horizontal track From the position versus time graph for when the cart is coasting (Be sure to include units) Print the graph and label on the graph the region where the cart is coasting. From the velocity versus time graph for region where the cart is coasting x- Print the graph and label on the graph where the cart is coasting Case 2 Cart moving down an incline plane From the position versus time graph for when the cart is accelerating down the incline Print the graph and label on the graph the region where the cart is accelerating down the incline From the velocity versus time graph for region 3 x- Print the graph and label on the graph where the cart is accelerating down the incline 11

5 Case 3 Cart moving up an incline plane From the position versus time graph for when the cart is given a push up the incline Print the graph and label on the graph the region where the cart is a push up the incline From the velocity versus time graph for region 3 x- Print the graph and label on the graph where the cart is a push up the incline Case 4 Determining the acceleration of gravity x = y = = For case 2 (cart moving down the incline) For case 3 (cart is given a push up the incline) g =a/sin = g =a/sin = How does the value for g compare to the known value of g? 12

How do you roll? Fig. 1 - Capstone screen showing graph areas and menus

How do you roll? Fig. 1 - Capstone screen showing graph areas and menus How do you roll? Purpose: Observe and compare the motion of a cart rolling down hill versus a cart rolling up hill. Develop a mathematical model of the position versus time and velocity versus time for