A projectile is fired with an initial velocity (v0) at an angle (θ) with the horizontal plane. Find an expression for the range (R). The data are given in the table below. Use the data in the table to create one or more dimensional plots (e.g., launch speed on the abscissa and range on the ordinate). From these plots, answer the following questions.
Launch Angle (U) [°] | Launch Speed (vQj [m/s] | Measured Range (R) [m] |
4 | 70 | 73 |
50 | 50 | 230 |
3 | 50 | 30 |
45 | 18 | 32 |
37 | 27 | 75 |
35 | 60 | 325 |
22 | 8 | 4.4 |
10 | 30 | 34 |
88 | 100 | 77 |
45 | 45 | 210 |
(a) If the launch speed is 83 meters per second and the launch angle is 64 degrees, what is the range? You will likely find it difficult to provide a good estimate of the range, but do the best you can.
(b) Complete a dimensional analysis of this situation. In this case, you would assume that the important parameters are θ, v0, and R. Upon closer examination, however, it would seem that the range on Earth and on the moon would be different. This suggests that gravity is important, and that you should include g in the list of parameters. Finally, since it is not clear how to include θ, you could omit it and replace the velocity by vx and vz, where x is the distance downrange and z the height. You should use this information to determine dimensionless parameters. Also, you must decide how the lengths in R and g should appear. When you complete the analysis, you should find that these four parameters will be grouped into a single dimensionless ratio.
(c) Use the data from the table to calculate the numerical value of the ratio for each test. Note that νx = ν cos(θ) and that you can find a similar expression for vz. Insert these expressions into your dimensionless ratio.
(d) Assuming that you performed the dimensional analysis correctly, you should find that the ratio you obtained will always give the same value (at least nearly, within test-totest error). Calculate the average value of the tests, and if it is nearly an integer, use the integer value.
(e) Finally, set this ratio equal to this integer, and then solve for the range R. Write your final equation for the range (i.e., R = xxxxx). Now using this equation, answer question (a) again.
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