Lab #4: 2-Dimensional Kinematics. Projectile Motion

Transcription

1 Lab #4: -Dimensional Kinematics Projectile Motion A medieval trebuchet b Kolderer, c Introduction: In medieval das, people had a ver practical knowledge of projectile motion. The ma not have understood the exact trajector that a projectile would take, but b practice the could place a projectile on a target consistentl from a distance of well over 00 ards. During a long siege of a castle, it was not uncommon to hurl bodies of animals (and es, captives back into the besieged castle s water suppl (an earl form of biological warfare. Similarl, a modern da hunter does not need to know the actual path that a bullet takes to a target in order to hit the target. A sharpshooter, however, does know the path and can make adjustments in the aiming in order to hit a target at man different ranges. In this lab, ou will become a sharpshooter of sorts. You will use the equations of motion to predict the path of a projectile and hit a target. Neglecting frictional forces, such as air resistance, an object projected from a launcher undergoes a motion that is the simple vector combination of uniform velocit in the horizontal direction and uniform acceleration in the vertical direction. For a projectile launched with a speed, v(0, at an angle Θ with respect to the positive x axis, it can be shown that the trajector caused b such a combination predicts a parabolic shape. The following kinematic equations describe this motion: Horizontal Motion: Vertical Motion: = x(0 vx (0 t Eq 1 ( t = (0 v (0 t 1 a t Eq v = vx(0 v ( t = v (0 a t Eq 3 v ( t = v (0 a ( ( t (0 Eq 4 Lab#3 D Kinematics

2 Where v (0 and v x (0 are the initial vertical and horizontal components of the velocit respectivel. Notice that Equations 1 and have a common variable, t. Equation 1 predicts the x coordinate in terms of the parameter t, Equation predicts the coordinate in terms of the parameter, t. B combining these two equations, the dependenc upon the parameter, t, can be eliminated. Simpl solving Equation 1 for t and substituting it into Equation results in the following: ( (0 (0 1 x = v x a x vx(0 vx(0 Where: x = x(0 Which simplifies to: v(0 a ( x = (0 x x vx(0 ( vx(0 ( Eq 5 Furthermore, the components of the velocit can be written in terms of the original launch velocit as: v x ( 0 = v(0 cosθ v ( 0 = v(0 sin Θ These components, when combined with Equation 5 ield an equation for (x determined completel b v(0 and Θ (the initial launch speed and angle: a ( x = (0 ( tanθ x x ( v(0cos Θ ( Notice that Equations 5 and 6 describe the position of the object but the do not sa when (at what time the object has an particular position. Also, notice that the relationship between and t in Equation is quadratic (parabolic in t because the values for a, v (0, and (0 are constant. Similarl, in Equations 5 and 6, vertical position,, as a function of horizontal position, x, is quadratic (parabolic in? x because the values for a, v (0, and v x (t are also constant. The equation for (x represents the trajector of the projectile. If a, x(t, x(0, Θ, and (0 are known, then it should be possible to determine the speed at which the projectile was launched. Note: The famo us Range Equation for projectile motion is a special case of the derivation described above. It can onl be used when a projectile starts and lands at exactl the same vertical height. It also defines a coordinate axis for the trajector such that x(0 = 0 and (0 = 0. As an exercise, plot Equation 6 as vs. x for a variet of realistic values for Θ, v(0, and (0. What determines the shape of the curve, the x position of the maximum, and the height of the curve ( position of the maximum? Experimenting with the mathematics of a trajector can ield tremendous insight into projectile motion. Eq 6 Lab#3 D Kinematics

3 Lab #3: -Dimensional Kinematics Goals: Resolve velocit vectors into components. Determine the muzzle velocit of a projectile launcher. Predict the range of a projectile. Use Excel to analze the motion of a projectile. Equipment: Projectile Launcher Steel projectile Meter Stick or Tape Measure Table clamp Carbon paper Excel Paper Target Activit 1: Determining Launch Velocit (Tabletop to tabletop launch 1. Set up a projectile launcher at an arbitrar angle, other than 0 or 90 o. The launcher should be adjusted so that it projects the projectile onto the tabletop. The angle that ou set, q, is the angle that ou will use throughout the experiment. DO NOT point our launcher in the direction of the computer monitors!. Carefull measure the height from the tabletop to the launching position of the projectile. The manufacturer has placed a mark on the side of the launcher for the purpose of this measurement. This is the initial vertical position, (0. 3. The launcher has three ranges: each range is determined b a click in the spring launcher and is also marked on the side of the launcher. Be sure to use the second click (medium range setting. 4. Fire the launcher and have a lab partner note the approximate position that the projectile strikes the table. Tape a piece of paper to the tabletop and place a sheet of carbon paper (carbon side down on top of the taped paper. It is not necessar to tape the carbon paper to the table. 5. Fire for effect! Fire the launcher several times to obtain an average landing position. Estimate the center position of our pattern and measure the horizontal distance from this point to a point directl below the launching position of the projectile. This is the horizontal range, x. 6. Use the values for a, x(t, x(0, Θ, (x and (0 to calculate the speed, v(0, at which the projectile left the launcher. (Assume that x(0=0, and a = 9.8 m/s. 7. Record the values of Θ and the calculated launch speed, v(0, for use in the next activit as well as for the Lab#3 D Kinematics

4 Activit : Determining Projectile Range (Tabletop to floor launch In the previous activit, ou determined the speed and direction, hence the velocit, of a projectile being fired from our launcher. You will now use this information to predict the landing point of a projectile that is launched from the tabletop to the floor. 1. Turn our launcher so that it faces awa from the table and towards an open space on the floor. Measure the vertical distance from the launching point of the projectile (on the side of the launcher to the floor. This will be our new (0. Record this value of (0 for use in the Post Lab.. Use the value of v(0, (0, and Θ to determine the horizontal distance at which a target must be placed in order to be hit with our projectile. Activit 3: Range vs. Angle Theor predicts that, when a projectile starts and lands from the same vertical height, the maximum horizontal range should occur at 45 o. In addition, there should be two distinct angles (complementar angles of launch that would send a projectile to a particular range less than the maximum range. 1. Fire our projectile launcher (so that the projectile again lands on the tabletop at different angles from 0-90 at either 5 or 10 increments (depending on the amount of time available but include a data point for 45. Record data for range and launch angle in a table, such as: Launch Angle (degrees Range (meters. Using Excel, make a graph of Range vs. Angle. 3. Based upon our graph, does the maximum range occur at 45? If not, where does it occur? 4. From our graph, generate an estimated list of at least 5 pairs of angle measures that ield the same range values? Is each pair a set of complementar angles? Explain. 5. Make a record of our graph and/or data for use in the Post Lab. Lab#3 D Kinematics

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