OBJECTIVES Motion with Changing Speed Collect position, velocity, and acceleration data as a ball travels straight up and down. Analyze the position vs. time, velocity vs. time, and acceleration vs. time graphs. Determine the best fit equations for the position vs. time and velocity vs. time graphs. Determine the mean acceleration from the acceleration vs. time graph. INTRODUCTION It is an important case in kinematics to study the motion of objects that are moving with a constant or uniform acceleration. The most practical example of a uniform acceleration is that of the acceleration due to gravity. When objects fall solely under the influence of gravity, they are said to be in free-fall. They fall with an acceleration g, due to gravity. Near the Earth's surface, the acceleration due to gravity is relatively constant, with a given value of: g = 9.80 m/s 2 = 980 cm/s 2 = 32.2 ft/s 2 Generally, you will not find this value to be exact due to air resistance. Rather heavy/dense objects that only fall for a short period of time tend to fall with acceleration due to gravity where effects due to air resistance can be neglected. During this laboratory, you will make measurements of position vs. time for objects moving under the influence of gravity. From these measurements, the value of g can be experimentally determined and compared to the known value. Additionally, you will make comparisons to uniformly accelerated motion that is not due to gravity. An object in free fall will be studied. During the early part of the seventeenth century, Galileo experimentally examined the concept of acceleration. One of his goals was to learn more about freely falling objects. Unfortunately, his timing devices were not precise enough to allow him to study free fall directly. Therefore, he decided to limit the acceleration by using fluids, inclined planes, and pendulums. APPARATUS Computer Meter stick Logger Pro Colored pencils Vernier Motion Detector Rubber ball Motion with Changing Speed - Page 1
THEORY There are several basic equations to consider when an object undergoes onedimensional uniformly accelerated, vertical motion. The first deals with the velocity as a function of position, given by: y v= t Where, v [m/s] is the average velocity and Δy [m] is the change in the vertical distance traveled in the time interval Δt [s]. This equation form is general for any motion, not necessarily uniformly accelerated; even though the vertical direction is indicated. The next equation deals with the average velocity as it relates to uniformly accelerated motion. It is stated as: v +v v= 2 Where, v [m/s] is the final velocity and vo [m/s] is the initial velocity. The third equation states that the velocity of the object increases linearly with time. If the initial value of the object's velocity is zero, starting from rest, then its velocity at a later time will be: v o v= g or f = vo t g t Where, Δv [m/s] is the change in the velocity of the object (final velocity {vf} minus initial velocity {vo}) in time interval Δt [s] and g [m/s 2 ] is the average constant/uniform acceleration due to gravity of the object. By combining the above equations, the following final equation forms are generated: v y 2 = v 2 oy 2 g ( y yo) Which defines the velocity of an object by its initial velocity, constant acceleration, and position. Thus, when you toss a ball straight upward, the ball slows down until it reaches the top of its path. The ball then speeds up on its way back down. A graph of its velocity vs. time would show these changes. Is there a mathematical pattern to the changes in velocity? What is the accompanying pattern to the position vs. time graph? What would the acceleration vs. time graph look like? Motion with Changing Speed - Page 2
When an object undergoes free-fall (neglecting air resistance) under the influence of gravity (acceleration g) the previous equation becomes: y = y o + v oy t 1 2 g t2 Where, y [m] is the initial height from which the object falls, eventually reaching a final height of yo [m], and g [m/s 2 ] is the acceleration of gravity. If we assume that the object starts from rest (voy = 0) and reaches the ground (yo = 0), the previous equation becomes: 1 = 2 y 2 The downward direction is taken to be positive to avoid the use of negative signs. From this equation it is apparent that a simple measure of the time it takes an object to fall a known distance will result in the determination of the acceleration due to gravity. g t EXPERIMENTAL PROCEDURE 1. Connect the Vernier Motion Detector to the DIG/SONIC 1 channel of the interface. 2. Place the Motion Detector on the floor. 3. Open the file 06 Ball Toss from the _Physics with Vernier folder. Enlarge the position vs. time graph to full screen for now. You'll use the velocity vs. time graph later. 4. In this step, you will toss the rubber ball straight upward above the Motion Detector and let it fall back toward the Motion Detector. This step may require some practice. Hold the ball directly above and about 0.5 m from the Motion Detector. Click to begin data collection. You will notice a clicking sound from the Motion Detector. Wait one second and then toss the ball straight upward. Be sure to move your hands out of the way after you release it. A toss starting 0.5 m above the Motion Detector works well. 5. Examine the position vs. time graph. Repeat Step 4 if your position vs. time graph does not show an area of smoothly changing position. Check with Dr. Arts if you are not sure whether you need to repeat the data collection. Even after Dr. Arts approval of the overall form of your attempt, you must still verify that your graph has the correct mathematical function before moving forward. The procedure on the following page is used to verify the function of your graph. Motion with Changing Speed - Page 3
Position vs. Time - Graph Verification Procedure The form of your position vs. time graph should be that of a parabola (that form associated with a projectile). As such, mathematically, it should fit the curve of a quadratic equation.you need to verify this fit. You will do this by clicking and dragging the mouse across the portion of the position vs. time graph that is parabolic. o This will be the TOP part of the arc ONLY!! Be especially careful NOT to include the portion of the graph where the ball was in your hand! Click the Curve Fit button,, and select the Quadratic fit equation (Ax^2+Bx+C) from the list of general equations; then click. A list of the equation s coefficients (A, B, & C) will appear as numerical values in a list on the right of the Curve Fit box. Pay attention to the A coefficient value. If this value is not approximately -4.905 then you will need to redo your graph and recheck its quadratic fit. o Click cancel to remove the data box. Once you have a set of the three graphs (y vs. t, v vs. t, and a vs. t).with the correct form, you need to do a couple of things before they are printed: o You're going to want to select the portion of the graph that corresponds to the desired motion (likely there are parts before and after that are non-useful data). Once selected, you're going to use the ZOOM feature of Logger Pro to enlarge that region. The selected region must include only the "in hand", "up", "top", and "down" portions of the motion. DO NOT include the section at the end where the ball has hit the floor/stool/sensor/etc. Motion with Changing Speed - Page 4
Be sure to select the EXACT SAME region of time on each graph; this way a clear comparison can be made. For example, if the highlighted region on your y vs. t graph is from 1s - 3.2s, then the region you need to select on the v vs. t & a vs. t graphs must also be from 1s - 3.2s. o Then, you will need to enlarge each of the three graphs individually on the screen as large as possible by simply stretching the graph frame. The graphs you have recorded are fairly complex and it is important that they be large and clear in order to be able to identify different regions of each graph. o Individually display each graph on the screen and print it out in landscape mode; be sure to have your footer label so you get your own set of graphs from the printer. o Finally, DO NOT erase these three graphs...you will be completing some additional analysis with the shortly. COVER PAGE REPORT ITEMS (To be submitted and stapled in the order indicated below) (-5 points if this is not done properly) Lab Title Each lab group member s first and last name printed clearly Color Group Date DATA (worth up to 20 points) The zoomed-in, full-page print outs of each of the three data graphs: y vs. t, v vs. t, and a vs. t. DATA ANALYSIS (worth up to 40 points) For the position vs. time (y vs. t) graph & coding using the colored pencils: o Identify the region when the ball was being tossed but still in your hand o Identify the region where the sensor stopped seeing your hand and started detecting the ball s motion under the influence of gravity ONLY o Identify the region where the ball is in free fall upward o Identify the region where the ball is at its maximum height o Identify the region where the ball is in free fall downward Motion with Changing Speed - Page 5
Your descriptions here, aside from the general description indicated above, should highlight "WHY" the graph indicates what it does. It is not merely sufficient to say a graph "clearly shows a object moving upwards up from rest." You must describe why that particular graph indicates that type of motion (for example) based on the line's format and/or shape. The motion of an object in free fall is modeled by y = yo + v 0 t - ½ gt 2, where y is the vertical position, yo is the initial vertical height, v 0 is the initial velocity, t is time, and g is the acceleration due to gravity (9.8 m/s 2 ). This is a quadratic equation whose graph is a parabola. Your graph of position vs. time should be parabolic. To fit a quadratic equation to your data, click and drag the mouse across the portion of the position vs. time graph that is parabolic, highlighting the free-fall portion. Click the Curve Fit button, select Quadratic fit from the list of models and click. Examine the fit of the curve to your data and click to return to the main graph. Print a copy of this graph with the curve fit and data box indicated. Place it behind the original y vs. t graph for submission. Explain the relevance of the quadratic equation coefficients (A, B, & C) as they relate to the equation y = yo + v 0 t - ½ gt 2. This information is a compare and contrast of the values in the data box now on the graph in relation to the equation x = At 2 + Bt + C. This explanation must be both qualitative (what do the coefficients match up to and are the numerical values reasonable in the context of the experiment performed) and quantitative (for the coefficient associate with gravity only; as this is able to be compared to a known constant). For the velocity vs. time (v vs. t) graph & coding using the colored pencils {MATCH the same color to each section used for the position vs. time graph previously}: o Identify the region when the ball was being tossed but still in your hand o Identify the region where the sensor stopped seeing your hand and started detecting the ball s motion under the influence of gravity ONLY o Identify the region where the ball is in free fall upward o Identify the region where the ball is at its maximum height What was the velocity of the ball at the top of its motion? o Identify the region where the ball is in free fall downward Your descriptions here, aside from the general description indicated above, should highlight "WHY" the graph indicates what it does. It is not merely sufficient to say a graph "clearly shows a object moving upwards up from rest." You must describe why that particular graph indicates that type of motion (for example) based on the line's format and/or shape. The graph of velocity vs. time should be linear and related to the equation vf = vo - gt. To fit a line to this data, click and drag the mouse across the free-fall region of the motion. Click the Linear Fit button Motion with Changing Speed - Page 6
Print a copy of this graph with the linear fit and data box indicated. Place it behind the original v vs. t graph for submission. Explain the relevance of the slope coefficients (m & b) as they relate to the equation vf = vo - gt. This information is a compare and contrast of the values in the data box now on the graph in relation to the equation v = mt + b. This explanation must be both qualitative (what do the coefficients match up to and are the numerical values reasonable in the context of the experiment performed) and quantitative (for the coefficient associate with gravity only; as this is able to be compared to a known constant). For the acceleration vs. time (a vs. t) graph & coding using the colored pencils {MATCH the same color to each section used for the position vs. time & velocity vs. time graphs previously}:: o Identify the region when the ball was being tossed but still in your hand o Identify the region where the sensor stopped seeing your hand and started detecting the ball s motion under the influence of gravity ONLY o Identify the region where the ball is in free fall upward o Identify the region where the ball is at its maximum height What was the acceleration of the ball at the top of its motion? o Identify the region where the ball is in free fall downward Your descriptions here, aside from the general description indicated above, should highlight "WHY" the graph indicates what it does. It is not merely sufficient to say a graph "clearly shows a object moving upwards up from rest." You must describe why that particular graph indicates that type of motion (for example) based on the line's format and/or shape. The graph of acceleration vs. time should appear to be more or less constant. Click and drag the mouse across the free-fall section of the motion and click the Statistics button,. Print a copy of this graph with the data box indicated. Place it behind the original a vs. t graph for submission. How closely does the mean acceleration (from the value in the data box now on the graph) compare to the values of g found in the y vs. t and v vs. t analyses? What should the mean acceleration for the trip have been? How does the value compare as the ball was moving up, at its maximum height, and as it was falling back down? GRAPHS (worth up to 0 points) None Required GRAPH ANALYSIS (worth up to 0 points) None Required Motion with Changing Speed - Page 7
CONCLUSION (worth up to 20 points) See the Physics Laboratory Report Expectations document for detailed information related to each of the four questions indicated below. 1. What was the lab designed to show? 2. What were your results? 3. How do the results support (or not support) what the lab was supposed to show? 4. What are some reasons that the results were not perfect? QUESTIONS (worth up to 10 points) DO NOT forget to include the answers to any questions that were asked within the experimental procedure; if not already answered on the graph for which the question was asked. Motion with Changing Speed - Page 8