Projectile Motion THEORY. r s = s r. t + 1 r. a t 2 (1)

Transcription

1 Projectile Motion The purpose of this lab is to study the properties of projectile motion. From the motion of a steel ball projected horizontally, the initial velocity of the ball can be determined from the measured rane. For a iven initial velocity, the projectile rane will be measured for various initial anles, and also calculated by applyin the theory for motion with constant acceleration. For further backround information, refer to the sections in your textbook on projectile motion and motion with constant acceleration. THEORY For a iven initial velocity, v 0, and initial position, s 0,the position of a particle, s, as a function of time, underoin constant acceleration, a is iven by r s = s r 0 + v r 0 t + 1 r a t (1) This is a vector equation and can be broken up into its x, y, and z components. Since the motion is in a plane, we need only look at the x and y components. If we nelect air resistance, the acceleration in the y direction is -, due to ravity. The acceleration in the x direction is zero. Hence, the vector equation (1) becomes two scalar equations: x = x 0 + v 0x t () y = y 0 + v 0y t - 1 t (3) In terms of the anle θ, and the initial speed v o, the initial velocity components are v 0x = v 0 cosθ and v 0y = v 0 sin θ A. For the case in which θ = 0 (initial velocity is horizontal), Eqs. () and (3) become (4) If we eliminate t in Eqs.(5) we et y as a function of x. x = v 0 t and y = h - 1 t (5) x y = h (6) v o When the object hits the floor the x and y positions are x = R and y = 0. Hence, Eq. (6) becomes and solvin for v o we et R 0 = h v o 1 Projectile Motion

2 v 0 = R h for the case θ = 0 (7) B. Now consider the case in which θ 0 (initial velocity is not horizontal). If we solve Eq. () for t and substitute the result into Equation 3, (usin x o = 0 and y o = h) we et y = h + v 0y x - v 0x v x 0x (8) We can use Eqs. (4) to rewrite this in terms of quantities you will measure: y = h + xtan θ - x v 0 cos θ (9) Aain, the rane R is the value of x when y = 0; substitutin this ives h + R tan θ - R = 0 v 0 cos θ (10) This is a quadratic equation in R; solvin for R we et the rane equation, R = v 0 cosθ sin θ ± sin θ + h v 0 (11) APPARATUS Sprin un set-up Gun mounted on frame w/ protractor Hook collar w/ pointer Bench clamps Aluminum bar supports w/ win nut Short rod Sprin un ball Bar level Two-meterstick Plumb-bob Maskin tape Paper EXPERIMENT The experiment consists of measurin the rane R of a small ball fired from a sprin un at various anles from the vertical. The apparatus (see Fi. 1) allows both the anle of projection and the initial velocity to be varied. (Note: Do not load or fire the sprin un until its use has been described and demonstrated by your instructor) The anle of projection is measured with a protractor, and the initial velocity can be varied by adjustin the tension on the sprin with the adjustin knob on the back of the un. After the instructor has demonstrated the use of the un, load it and fire a few practice shots. Adjust the sprin tension so that the rane, when fired horizontally, is between one and two meters. Practice firin the un the same way each time to minimize the variation in initial velocity. Projectile Motion

3 Two-meterstick Adjust knob yo = h θ Lock knob Recordin Paper x = R y = 0 x = 0 y = 0 Fiure 1: The sprin un setup. PROCEDURE 1. Level the un usin the level provided. Tihten the knurled brass screw. Now adjust the pointer so that it indicates 0. Leave the un in this position and be sure all screws and clamps are tiht.. Use a plumb bob to locate the point on the floor directly under the point where the ball leaves the un. Mark this point on a piece of paper or tape fastened to the floor. Measure the heiht h of the ball from the floor. (Should you measure to the center of the ball or the bottom of it?) Note that this heiht chanes as the anle is varied. 3. Determine the approximate rane of the ball by firin the un a couple of times (place a backstop beyond the landin point to protect innocent passersby). In order to measure the position at which the ball strikes the floor, tape a sheet of paper to the floor such that the ball strikes the paper approximately in the center. 4. Fire at least ten shots; circle and number the impact positions on the sheet on the floor. If the positions are spread so much that they do not all hit the paper, check that everythin is tiht and that each time you use the same technique to project the ball. 5. Determine the rane, R, by measurin the distance from the spot under the un (x = 0 and y = 0) to the marks on the recordin sheet. Use statistical methods to find the mean rane and the standard deviation. 6. Dependence of Rane on Anle of Projection θ (v o fixed): 3 Projectile Motion

4 Before takin more data, vary the anle of launch and try to find the anle which ives the maximum rane. You may want to try predictin it first. Make just a couple of shots at each anle. Are you surprised? If so, read question 6 below. Repeat steps throuh 5 for at least three more anles, between 10 and 50. Be certain not to turn the sprin adjustin knob durin this procedure or you will chane the initial velocity. Be sure to measure the new initial position (both x o and y o ) as the anle chanes. 7. Dependence of Rane on Initial Velocity v o (θ fixed). a. Chane the initial velocity of the ball by adjustin the knob on the back of the un. Repeat Steps throuh 5 with the un in the horizontal position to obtain information for calculatin v o. Then chane the anle of launch to one of the anles used in Part 6 and aain repeat steps throuh 5. b. Chane the initial velocity aain and repeat step 7a, usin first θ = 0 and then the same non-zero anle θ for this initial velocity. ANALYSIS & QUESTIONS 1. If we choose the point directly below the end of the un as x = 0 and the floor level as y = 0, then x o = 0 and y o = h. Use this information and the mean rane measured for the horizontal shots to calculate the initial velocity (usin Eq. (7)) for each sprin tension settin; v o is assumed to be the same independent of anle of projection. Is this a ood assumption?. Usin the rules for error propaation (see handout Errors and the Treatment of Data), determine the standard deviation of v o from the measured standard deviation of R meas and the error in the measurement of h (see Eq. (7)). Show this calculation in your lab report. 3. Usin the appropriate initial velocity, v o, determined in question 1, calculate the rane, R,for all non zero anles. calc 4. Compare and discuss your values for R meas and R calc as a function of anle and as a function of initial velocity. Are the differences R meas - R calc consistent with your calculated statistical and measurement errors? What factors or uncertainties associated with the apparatus contribute to the spread in the measured quantities? Can you estimate the uncertainty in the rane resultin from any of these factors? 5. Based on your data (Step 6 of the procedure), for approximately what anle of projection was the rane reatest? Explain qualitatively why this is so (see Question 6). 6. Show that for a launch from the floor (h = 0), Eq. (11) reduces to R = 0 or R = v 0 sin θ, 4 Projectile Motion

5 the rane equation in your text. 7. How miht you directly measure the initial velocity of the ball? What additional apparatus would you need? 8. In the rane equation, Eq. (11) of the Theory section, there is a ± sin. Which sin do you use? What, if any, is the sinificance of the other sin? 9. Ideally what mathematical curve is the trajectory? 10. Describe two examples of projectile motion which you have observed or experienced outside of the physics lab. 5 Projectile Motion

6 DATA SHEET PROJECTILE MOTION TA [ ] NAME: INSTRUCTOR: PARTNER: SECTION: DATE: Dependence of rane on projection anle, for fixed initial velocity. Horizontal: θ = 0 h = R meas = σ(r meas ) = v o = σ(v o ) = # θ (de) h (meters) R meas σ(r meas ) R calc σ(r calc ) R meas -R calc Dependence of rane on initial velocity at a fixed anle. Recopy appropriate data from above for #1. # θ h R meas σ(r meas ) v o σ(v o ) R calc σ(r calc ) R meas -R calc Projectile Motion