Lab 3 - Projectile Motion

PHYS 31, SCU Physics Dept., Winter 2022 (online) Name: Lab Partners: Lab 3: Projectile Motion In this lab you will study two dimensional projectile motion of an object in free fall - that is, an object launched into the air that falls back to the ground under the influence of gravity only. To describe the trajectory of the projectile, we will use a coordinate system where the positive y-axis is vertically upward and the x-axis is horizontal and in the direction of the initial launch velocity (see Figure 1). We will we assume that the gravitational acceleration ( g = 9.81 m/s 2 ) is constant, so that a x = 0 and a y = − g , and we will ignore air resistance. The equations of motion in the x and y directions for such a projectile launched with initial velocity ⃗v o = v ox ˆ x + v oy ˆ y at an angle θ are: x ( t ) = x o + v ox t (1) v x ( t ) = v ox (2) y ( t ) = y o + v oy t − 1 2 gt 2 (3) v y ( t ) = v oy − gt (4) where t is time. The x and y components of the initial velocity ⃗v o are v ox = v o cos ( θ ) and v oy = v o sin ( θ ) for the geometry shown in Fig. 1a. Remember that v x = v ox here because there is no acceleration in the x direction. Figure 1b shows the path of a projectile with initial speed v o , launch angle θ and initial height H relative to the ground. Using Eqs. 1- 4 you should be able to show (optional!) that the projectile’s path, i.e. its vertical position y as a function of its horizontal position x is described by: y ( x ) = H + tan ( θ ) x − gx 2 2( v o cos ( θ )) 2 (5) where x = x ( t ) and g = 9 . 81 m/s 2 . Notice that Eq. 5 describes a parabola! Last but not least, notice that the range R of the projectile, i.e. , the total horizontal distance traveled by the projectile in the total flight time t tot is: R = v x 0 t tot = v 0 cos ( θ ) t tot (6) θ ~ v o cos ( ✓ ) ~ v o sin ( ✓ ) ~ v o (a) θ R y max carbon paper H ~ v o Boom! (b) Figure 1: (a) Vector components of the initial velocity ⃗ v o . (b) A projectile is launched at an angle θ off a platform with initial height H relative to the ground. The projectile strikes the ground a horizontal distance R away from the launch point. 1 Lam Reese Monica, Durgeshwaran

Experimental (Simulation) Set-up In this lab you will use a projectile launcher applet provided by PhET that can be found here (click the “Lab” icon). The user interface is shown in Figure 2. The projectile launcher can be positioned at different heights by clicking-and-dragging the platform, and can launch the ball with different initial velocities (magnitude and direction). The protractor on the side of the launcher allows you to set the desired launch angle, ranging from 0 ◦ to 90 ◦ . The built-in tape measure widget will be used to measure various distances. You will need an external stop-watch to record the amount of time t tot the projectile is in the air. You can use your cellphone’s stopwatch since it’s quick and easy to use. With this virtual equipment, you will determine the initial launch velocity ⃗ v o , an experimental value for | ⃗g | , and the launch angle θ that provides the longest range when the projectile is launched from above ground level. Figure 2: Use interface for PhET Projectile Motion applet. Part 1: Finding the Initial Launch Velocity You will use two different methods to determine (that is, confirm) the initial speed of the ball when launched from 5 meters above ground level, with an initial velocity | ⃗ v o | at an angle of 40 degrees. 1. Go to the PHET simulation and click the “Lab” icon. Position the launcher to 40 ◦ and 5 m above the ground. Set the initial velocity | ⃗ v o | to 30 m/s. Use the built-in feature of the simulator to zoom out enough to get a complete view of the ball trajectory (top left icon). In order to minimize random errors, it is very important that all of your measurements are made several times. Each lab group member must make at least two sets of time and range measurements and then share them with the group. Use your phone stopwatch to measure time and use the built-in tape measure widget to measure range. Use the eraser button to reset the simulation window before starting each trial. 2

5. Getting the initial speed v o from time measurements: Now use Eq. 1 to determine the initial launch speed of the projectile. v o = 6. Compare your two results for v o with the value you used in the simulation. What were the biggest sources of error in your measurements? Part 2: Maximum Height and the Acceleration Due to Gravity on Earth The goal of this part of the lab is to experimentally determine the magnitude of the acceleration due to gravity, g , near the surface of the Earth. As before, each student will collect at least two data points and share them with the rest of the group so that all rows of the data table below are filled. Eqs. 1-4 can be used to write the maximum height y max reached by a projectile as: y max = v 2 o sin 2 ( θ ) 2 g (7) where v o is the launch speed, θ is the launch angle, and g is the acceleration due to gravity. Now you will use this equation and maximum height measurements to determine an experimental value for g . Notice the simple linear relationship (i.e. y = mx+b) between y max and v 2 o in Eq. 7. 1. Equation 7 only holds when the projectile launch and landing heights are the same, so set H = 0 m in the simulation. Keep the launch angle fix to θ = 40 ◦ . Launch the ball and measure its maximum height above the ground (use the measure tape) for each of the initial speeds in the table below. Record your maximum height values, calculate the corresponding v 2 o values and add the proper units at the top of each column. v 0 y max ± ∆ y max v 2 o 15 16 17 18 19 20 21 22 4 22.47m/s They are different because the v0 we got from the time measurements is only the x-component of velocity instead of the initial velocity vector. The value for initial velocity we calculated had only 0.9% difference than the true initial velocity. I think the biggest sources of error were our time and range measurements and rounding inconsistencies.

2. Plot y max as a function of v 2 o . Include error bars ± y max on your graph. Draw a line of best fit through your data (with error bars) and calculate its slope. Include units! slope = 3. Use Eq. 7 to derive the expression for g as a function of the slope of your graph. 4. Use your slope to calculate g and compare your result with accepted value of 9.81 m/s 2 . Comment on how your error bars can (or cannot) account for any discrepancy. Part 3: Launch Angle and Range 1. If a projectile is launched from ground level, the launch angle of 45 ◦ would provide the largest possible range. Predict if the launch angle corresponding to maximum range should be larger or smaller than 45 ◦ if the projectile is launched from above the ground level ( H > 0). State your reasoning. 5 0.0211 (m/m/s^2 s in 2 ( θ ) 2 *slope g = g = 9.7809 I think it should be less than 45 degrees because when the height is increased, there is more vertical distance to cover and therefore more time for the object to cover horizontal distance. When theta is lower then, the x component of the initial velocity will be increased therefore making it go further. Error bars can account for discrepancies because they will affect the slope and therefore the calculated value of g. g= sin^2(40)/(2*0.0211)