In the above diagram, a bail bearing starts from rest at height h above. a lab t
ID: 1709078 • Letter: I
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In the above diagram, a bail bearing starts from rest at height h above. a lab table. It rolls along a steel track and leaves the track with a horizontal speed V0. The marble follows a parabolic trajectory to the floor where it lands a horizontal distance R away from the edge of the table. Use your kinematics equations and error propagation formulas to answer the following questions: Derive a formula for the speed of an object dropped from a height h in a vacuum. (This is equivalent to the speed V, in our experiment.) Show algebraically that the time t that the ball bearing takes to hit the floor after it leaves the track is given by the expression t = 2H / g , where g is the acceleration of gravity. Show that the range R of the ball bearing may be written at R = 2 hN. Given that the error in measuring h and H are delta H, respectively, show that the propagated error in the formula for R in question 3 is given by delta R / r = (delta h / 2h)2 + (delta H / 2H)2 (You may wish to review Pre·Lab 01: Primer on Experimental Errors;)Explanation / Answer
3 The range can be found by D=RT distance(or range) = rate times time. The horizontal speed doesn't change since gravity is perpendicular to it. We have the time from 2. To find the rate, we use conservation of energy. The potential energy at h is converted to kinetic energy at the corner of the lab table. mgh = 1/2 mv^2 v^2 = 2gh v = SQRT(2gh) Range = R = vt = SQRT(2gh)*SQRT(2H/g) = SQRT(4hH)= 2SQRT(hH) The error in h has nothing to do with the error in H. When this is the case we need to be thoughtful in how we combine the errors. Should we add them? For example should we assume that if we measure h shorter than it is that we also measure H shorter than it is. This gives worst case error. On the other hand we could assume that if we measure h shorter than it really is, we would measure H longer than it is. This gives best case error. Typically errors add like the sides of a right triangle with the total error growing like the hypotenuse. So errors do get larger, but not linearly. This is why you have the errors added like in the Pythagorian theorm. If the formula depended upon h and H to the first power there would be no denominators inside the square root. Since error depends on each h and H to the one half power, each term within is multiplied by 1/2 before it is squared.
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