Acceleration Team: Part I. Uniformly Accelerated Motion Acceleration is the rate of change of velocity with respect to time. In this experiment, you will study a very important class of motion called uniformly-accelerated motion. Uniform acceleration means that the acceleration is constant independent of time and thus the velocity changes at a constant rate. The motion of an object (near the earth s surface) due to gravity is the classic example of uniformly accelerated motion. If you drop any object, then its velocity will increase by the same amount (9.8 m/s) during each one-second interval of time. Galileo figured out the physics of uniformly-accelerated motion by studying the motion of a bronze ball rolling down a wooden ramp. You we study the motion of a glider coasting down a tilted air track. A. The Big Four: t, x, v, a The subject of kinematics is concerned with the description of how matter moves through space and time. The four quantities, time t, position x, velocity v, and acceleration a, are the basic descriptors of any kind of motion of a particle moving in one spatial dimension. They are the stars of the kinema. The variables describing space (x) and time (t) are the fundamental kinematic entities. The other two (v and a) are derived from these spatial and temporal properties via the relations v dx/dt and a dv/dt. Lets measure t, x, v, a. First make sure that the track is level. The acceleration of the glider on a horizontal air track is constant, but its value (a = 0) is not very interesting. In order to have a 0, you must tilt the track. Place two wooden blocks under the leg of the track near the end where the motion sensor is located. Release the glider at the top of the track and record its motion using the motion sensor. [Click on Logger Pro and open file Changing Velocity 2]. The graph window displays x, v, and a as a function of time t. Your graphs should have the following overall appearance: good data region x parabola t v linear a t 1 constant t 2 t t
Focus on the good data region of the graphs where the acceleration is constant. To find this region, look for that part of the graphs where the x, v, a curves take on smooth well-defined shapes: x = parabola, v = linear (sloping line), a = constant (flat line). In the bad data region, there are erratic variations in the curve. Here the acceleration is changing because the glider is experiencing forces other than gravity, such as your hand pushing the glider or the glider hitting the bumper. The irregular graphs may also be due to the motion sensor not seeing the glider. Identify two times (t 1 and t 2 ) within the good (constant a) region and record their values in the table below. Also record the values of the position (x 1 and x 2 ) and the velocity (v 1, v 2 ) at these two times. You can read the values of t, x, v, a on the graph by using the Examine Icon [x=?]. You can also read the values from the table window on the right side of the screen. Print the graph and include it in your report. Table of t, x, v Values t 1 = x 1 = v 1 = t 2 = x 2 = v 2 = On your printed graph, label the points 1 and 2 and write the corresponding values of t 1, t 2, x 1, x 2, v 1, and v 2. B. Acceleration Average a and Instantaneous a From your table of t, x, v values, calculate the average acceleration of the glider between t 1 and t 2 : v 2 v 1 ------------- = m/s 2. t 2 t 1 The instantaneous acceleration at time t is a dv/dt. Graphically speaking, the derivative dv/dt is the slope of the tangent line at the point (t, v) on the v(t) curve. Click on the Slope Icon [m=?] and find the values of dv/dt at four different points on your v(t) curve: dv/dt =,,,. Can you conclude that the acceleration of the glider is constant? Explain. 2
Kinematic Relations Compute the value of the acceleration (assumed constant) from your measured values of t 1, t 2, x 1, x 2 and the kinematic equation that relates t, x, and a. Show your calculation. a = m/s 2. Compute the value of the acceleration (assumed constant) from your measured values of x 1, x 2, v 1, v 2 and the kinematic relation that relates x, v, and a. Show your calculation. a = m/s 2. Compare these values of a with Δv/Δt and dv/dt above. Are all your values of a within 10% of each other? The Best Value of a There are several sources of error that can cause the acceleration of the glider to fluctuate over time. These errors include a bumpy track, dirt on track, a bent glider, dirt on glider, sliding friction, air friction, and the fact that the motion sensor approximates the continuity of motion the smooth flow of time by collecting and analyzing data in discrete time steps. The best value of the acceleration can be obtained by utilizing the information contained in all the data points, not just t 1 and t 2. This global analysis helps to smooth out the local fluctuations, thereby decreasing the effect of errors. One method is to average over the a(t) data. Over each ten-second time interval, the motion sensor collects one hundred values of a. This large amount of data stores valuable information on the motion that you can readily extract. Highlight the good data region of your a(t) graph. Click on the statistics icon [STATS]. In addition to computing the average value, the statistics method also gives the standard deviation away from the average value. The standard deviation provides a precise measure of the uncertainty in the average value. Report the average value of a(t) plus-or-minus the uncertainty. Average of a(t) = ± m/s 2. The second method is to find the slope of the best-fit line through the v(t) data. This method is based on the fact that acceleration is the rate of change of v(t). Highlight the good data region of your v(t) graph. Click on the curve-fit icon [f(x)=?] and perform a linear fit to find the best-fit line through the v-t data. The slope of this best-fit line is the best value of a. Slope of v(t) = m/s 2. 3
Natural gee units Since we live in the gravitational field of the earth and feel its strength daily, it is convenient to adopt the special number g = 9.8 m/s 2 as the natural unit of acceleration and express all other accelerations relative to g. An acceleration of 4.9 m/s 2 is equal to ½ g. Report the best value of the acceleration of your glider in g-units. a = g. C. Velocity From your table of t, x, v values, calculate the average velocity of the glider between t 1 and t 2 : x 2 x 1 ------------- = m/s. t 2 t 1 Is this value of v(average) exactly halfway between your measured values of v 1 and v 2 in the table? Show your calculation of (v 1 +v 2 )/2. ½ (v 1 + v 2 ) = m/s. The instantaneous velocity at time t is v dx/dt. Graphically speaking, the derivative dx/dt is the slope of the tangent line at the point (t, x) of the x(t) curve. Click on the Slope Icon [m=?] and find the values of dx/dt at four different points on your x(t) curve: t (s) dx/dt (m/s) What is the rate at which the slope is changing? i.e. calculate the change in dx/dt over the corresponding change in t. This double change (second derivative) in x with respect to t is written as d 2 x/dt 2. d 2 x/dt 2 = m/s 2. In theory, how is d 2 x/dt 2 related to a? How does your experimental value of d 2 x/dt 2 compare to your values of a measured previously? 4
D. Displacement From your table of t, x, v values, calculate the displacement of the glider between t 1 and t 2 : x 1 x 2 = m. The displacement has an elegant geometric interpretation: x 1 x 2 = Area under the v(t) curve between t 1 and t 2. The connection between area and displacement follows by writing v = dx/dt as dx = vdt. The total displacement Δx is obtained by adding up the tiny displacements dx = vdt: Δx = vdt. The quantity vdt is the area of the tiny rectangle of base dt and height v. This means Δx = tiny areas. Therefore Δx = total area under v(t). In Calculus, finding areas is called integration. Integrating (summing) both sides of dx = vdt gives Δx = vdt. The sum sign is written as the integral sign. Highlight the region of your v(t) curve between t 1 and t 2. Click on the area icon [π] and report the area: Area under v(t) = m. To check the computer result, calculate this area yourself by using the geometric formulae: area of rectangle = base x height and area of triangle = ½ base x height. Show this calculation on your printed graph. Area under v(t) = m. Compare your value of x 1 x 2 with your value of Area under v(t). Remember: Compare means Find the percent difference. Part II. The Physics of Free Fall Consider an object of mass m that is released from rest near the surface of the earth. After a time t, the object has fallen a distance y and is moving with velocity v. The free-fall equations relating y, t, and v are where g = 9.8 m/s 2 is independent of m. y = ½ gt 2, v = gt, v 2 = 2gy, In this experiment, you will test these important properties of free-fall motion by studying the motion of a glider on a tilted air track. Strictly speaking, free fall refers to the vertical motion of a body that is free of all forces except the force of gravity. A body moving on a friction-free inclined track is falling freely 5
along the direction of the track. It is non-vertical free fall motion. The track simply changes the direction of the fall from vertical to diagonal. This diagonal free fall is a slowed-down and thus easier-to-measure version of the vertical free fall. The acceleration along the track is the diagonal component of the vertical g. This acceleration depends on the angle of incline. It ranges from 0 m/s 2 at 0 o (horizontal track) to 9.8 m/s 2 at 90 o (vertical track). In other words, the track merely dilutes gravity. A frictionless inclined plane is a gravity diluter. We will denote the acceleration a of the glider falling down the track as the diluted value of the full-strength g: a = g(diluted). A. Experimental Test of the Squared Relation d t 2 In theory, the distance d traversed by the glider along the track is proportional to the square of the time elapsed (after starting from rest). This means that if you double the time, t 2t, then the distance will quadruple, d 4d. More specifically, if it takes time t 1 to move distance d 1 and time t 2 to move distance d 2, then the proportionality d t 2 implies the equality d 2 /d 1 = (t 2 /t 1 ) 2. Hence if t 2 = 2t 1, then d 2 = 4d 1. Tilt the track by placing two blocks under the end of the track. Use a stopwatch not the motion sensor to measure the time it takes the glider, starting from rest, to move a distance of 25 cm down the track. Repeat three more times to find an average time. Next measure the time it takes to move a distance of 100 cm. Record your measured times in the following table: Average Time t (d = 25 cm) t (d=100 cm) (s) (s) Are your experimental results consistent with the theoretical relation d t 2? Explain carefully by constructing ratios. Calculate the value of the acceleration a = g(diluted) of the glider along the track direction from your measured values of d and t. Show your calculation. g (diluted) = m/s 2. 6
B. Experimental Test of v 2 H Physics Fact: The speed v of an object, starting from rest and falling down the frictionless surface of an inclined plane, depends only on the vertical height H of the fall and not the length of the incline. Furthermore, the square of the velocity is proportional to the height: v 2 H. This squared relation implies that the speed will double if the height quadruples. H v 4H 2v Since you are testing the proportionality, v 2 H, and not the equality v 2 = 2gH, you only need to study how v depends on the number of blocks that you stack vertically to elevate the track. The height H can be measured in dimensionless units, simply as the number of blocks. Place one block (H = 1) under the motion-sensor end of the track. Position the glider at the point that is 20 cm away from the sensor. Release the glider from rest and measure its velocity v (using the sensor) when it is 100 cm away from the sensor. Repeat three more times to find an average velocity. Now quadruple the height by placing four blocks (H = 4) under the end. Once again, release the glider at 20 cm and measure its velocity at 100 cm Average Velocity v (H = 1) v (H = 4) (cm/s) (cm/s) Do your experimental results support the theoretical relation v 2 H? Explain carefully by constructing ratios. 7
C. Experimental Test of the Universality of g One of the deepest facts of Nature is that the acceleration of an object due to gravity does not depend on the size, shape, composition, or mass of the object. Use two blocks to incline the track. Use the motion sensor to record the motion of the glider as it falls freely down the track. Find the acceleration of the glider by averaging the a versus t data. Remember to carefully select the region of good data on the graph before you analyze the data. To find the average of the a(t) values, click on the statistics icon [STATS]. Report your results in the table below. Include the uncertainty (standard deviation) in each a and the range of each a. For example, if the statistical analysis of the acceleration data gives the average value 0.347 m/s 2 and the standard deviation 0.021 m/s 2, then you would report your measured value of acceleration to be 0.35 ± 0.02 m/s 2, or 35 ± 2 cm/s 2. The range of this a is 33 37 cm/s 2. Add two weights or metal donuts (one on each side of the glider) and measure the acceleration. Add four weights (two on each side) and measure the acceleration. m (kg) a ± uncertainty (cm/s 2 ) Range of a (cm/s 2 ) ± ± ± Do your experimental results support the deep principle that g, or equivalently g(diluted), is independent of mass? Remember that your measured values of a are equal to g(diluted). So the big question is: Are your three measured values of a equal? When are two experimental values equal? To answer this question, the role of uncertainty is vital. A measured value such as 15 ± 2 is really a range of numbers 13 17. Two experimental values are equal if and only if their ranges overlap. Suppose you are given two rods (A and B) and measure their lengths to be L A = 15 ± 2 cm and L B = 18 ± 3 cm. Since the two ranges overlap, 13 17 and 15 21, you can conclude that these two rods are equal in length. A range diagram provides an excellent visual display of the experimental values of measured quantities. The following range diagram for L A and L B clearly exhibits the amount of overlap: L A L B 13 14 15 16 17 18 19 20 21 cm 8
Now you can rigorously answer the important question: Do your measured values of a provide an experimental proof of the deep principle that the acceleration of gravity is independent of mass? Justify your answer by plotting your three values of a on the following range diagram: a 1 a 12 a 3 cm/s 2 Comments: Einstein, Curved Space, Black Holes, Warp Drive In a gravitational field, all bodies fall with the same acceleration. We have said this is a deep law of nature. Indeed, Einstein used this law as the basis for his general theory of relativity. All bodies fall in the same way because they are merely coasting along the same downhill contours of the curved space that they happen to occupy. Einstein s field equations tell you precisely how to calculate the curvature of four-dimensional space-time. Gravity is not a force it is the shape of space. The idea that gravity is curvature is the basis for bending light, gravity waves, black holes, and wormholes. In essence, your experimental proof that g is independent of m is a proof of the existence of black holes and gravity waves! When warp drive is invented, you will appreciate that it is a consequence of the universality of g. 9
Part III. Designing a Diluted-Gravity System In vertical free fall, an object released from rest moves about 45 m in 3.0 s. You need to slow this motion dilute gravity so that the object only moves 1.5 m in 3.0 s. Your goal is to find how much the track needs to be tilted to achieve this slowed-down motion. First work out the theory and then perform the experiment. Theory Architecture Diagram. H = height of blocks. L = distance between track legs. θ = angle of incline. track leg glider H L blocks θ leg table Acceleration Diagram. g = full strength gravity. a = diluted gravity. a g θ θ The height H of the blocks determines the acceleration a of the glider. Derive the algebraic relationship between H and a. The constants L and g will also appear in this relation. Hint: apply Sin Opp / Hyp to the two right triangles that contain θ in the above diagrams. Let t denote the time it takes the glider to move the distance d along the track. Derive the theoretical equation that gives H as a function of t, d, L, and g. H =. 10
The Design Specs say d = 1.5 m and t = 3.0 s. Measure the value of L. Note that L is not the length of the track (see Architecture Diagram). You know the value of g. Plug these numbers into your theoretical equation to compute the value of H. Show your calculation. H = cm. Experiment Again, make sure that the track is level. Raise the end of the track by the height H predicted above. To achieve the precise value of this height (within tenths of a centimeter), you will most likely need to place one or more of the thin square metal plates on top of the wooden blocks. Measure the thickness of the plate with the Vernier caliper. Release the glider from rest. Use a stopwatch to measure the time t it takes the glider to move 1.5 m along the track. Repeat this measurement five times. List your five values of t below and compute the average time and the uncertainty in the time. Estimate the uncertainty (deviation from average) from the spread in your five values of time: Uncertainty = (t max t min )/2. t (s) t = ± s. Compare this measured value of time with the design goal of t = 3.0 s. What is the percent difference? Clearly show whether or not the theoretical time of 3.0 seconds falls within your experimental range? If not, what source(s) of error could account for the discrepancy? 11