# Shooting a Balloon Lab

 Materials: Computer and School Network Time Allotment: 4 Class Days

### Introduction:

This activity simulates an archery contest in which you are challenged to pop a target balloon from a great distance. In the process you will learn about projectile motion. You will investigate the advantages of looking at a problem from more than one perspective. In particular, you will approach this task using two different coordinate systems: polar coordinates (Task 1) and Cartesian coordinates (Task 2).

1. Log on to the school server in the usual manner.
2. Open the Physics Explorer application (One Body model) by double-clicking on its icon; it is found in the directory Science-Math/Science Apps/Physics.
3. Choose Open... from the File menu and navigate to the Unit 3 folder in the Physics 163 folder.
4. Open the file titled Shooting a Balloon. Check that the values appear as below.

### Task 1: Set Angle and Shoot

You have a custom-built crossbow for this competition. It uses a plumb bob to read the launch angle. You have also tested the relationship between the bow extension and the launch velocity of the arrows. Using this information, you've constructed, calibrated and attached a meter to the cranking mechanism. The meter indicates the launch velocity which will be achieved for any given bow extension.

You need to find the correct angle(s) to the horizontal at which the bow must be set in order to hit the target. The launch velocity is fixed at 60 m/s.

1. Click on the "Set Target: Air" button in the Control Panel window to mark the target (the target is the samll crosshair).

2. Use the RULER tool in the Model window to find the target position (x and y) relative to the firing position. Note: Measure from the center of the arrow (circle) in its starting position first across horizontally and then up vertically. These two measurements will yield the horizontal and vertical distance from the arrow to the target balloon.
• Horizontal Distance (x) = ______________
• Vertical Distance (y) = ______________

Verify this position using an approximated reading off the "Target Screen" grid.

3. Use the Angle input in the lower-left window to vary the angle of the bow. Vary the angle and fire arrows until you find the correct angle to hit the target. The speed of the arrow is set at 60 m/s.
• Angle = ______________
• How many seconds did it take for the arrow to reach the target?
• For the same speed (60 m/s), is there more than one correct angle? ________ If so, record the different possibilities you discovered.

### Task 2: Set Velocity and Shoot

A representative of Renaisance Industries, Inc. comes by with a new crossbow based on a long-lost design by Leonardo DaVinci. It uses two bows at right angles to each other, allowing you to separately control the vertical and horizontal velocity components of the arrow. The arrow is on a pivot and it is automatically aimed depending upon the extension of the two bows. For instance, if the vertical bow is not etended at all (vy = 0 m/s), the force from the horizontal bow turns the arrow track completely to the horizontal. If both bows are extended equal amounts, then the arrow is directed at a 45-degree angle to the horizontal.

In Task 1, we were concerned with the polar coordinate description of the arrow's velocity - i.e., the speed and angle. Now the Renaisance representative convinces you that using this bizarre contraption will enable you to accurately predict the path of the arrow more easily using cartesian corrdinates - i.e., x-velocity and y-velocity. We have discussed earlier in our course that perpendicular components of motion are independent of each other. The x-components of motion have no effect upon the y-components of motion, and vice versa. In this task, you will utilize cartesian coordinate system and the principle of the independence of x- and y-components to ultimately strike the target.

1. To begin, click on the "Task 2" button. The lower-left window should now have the following appearance.

The controls in the Task 2 window allow you to input the x- and y-components of the velocity independently. (A button and two output boxes provide you with the calculated angle and arrow speed based on these components.)

2. Click on the "Set Target: Ground" button.
1. Set the x-velocity vx to 0 and the y-velocity vy to any value up to 60 m/s; record the y-velocity value below. Calculate how long it should take for the arrow to go up and come back down to the target (at ground level) if given this initial y-velocity (viy). HINT: vy at the peak of the trajectory is 0 m/s. So using a kinematic equatin (vi = vf + a*t), an equation for finding the time to reach the peak of the trajectory (tup) can be written.

g*tup + viy = 0

where g = -9.8 m/s/s (the acceleration of gravity)

The total time of flight (up and down) is twice the time to travel up.

Initial y-velocity (viy) = _____________ m/s

Calculated total time of flight: ____________ Show your work below.

Fire the arrow to check your prediction.

2. What x-velocity is necessary for the arrow to traverse the horizontal distance (as measured in Task 1) to the target during the time you calculated above? Use a kinematic equation to calculate the required x-velocity to cover the x-distance in the given amount of time with an x-acceleration of 0 m/s/s. PSYW

3. Enter this x-velocity and fire the arrow. How close were you to hitting the target balloon? _______ If you miss, then recalculate/adjust your values of vx until you successfully hit the target balloon. Use good calculations (refer to steps a and b) rather than a trial-and-error method.

3. Repeat step 2 for several different initial y-velocities (ranging from 20 m/s to 60 m/s). Calculate the time of flight and the required x-velocity for each chosen y-velocity. Run the simulation to check the results of your calculations. (For your own benefit, avoid replacing a cacluation approach by a trial-and error method.) Record your results and the show calculations in the table below:

 Initial viy (m/s) Calculated time of flight (s) Initial vx (m/s) Calculations/Work: _________ ___________ _________ ____________________ _________ ___________ ________ ____________________ _________ ___________ _________ ____________________ _________ ___________ _________ ____________________

4. Now, click on the "Set Target: Air" button and set the vy value between 50 and 60 m/s. Try to use the same kind of reasoning (as in steps 2 and 3 above) to hit an elevated target. First calculate the time of flight to the target based on the value of vy only. There are two possibilities for how to hit it: you could hit the target balloon on the way up or on the way down. There are also at least two posibilities for determining the time to get to this point.
• You could fire the arrow vertically and, using the "Single Step" button, measure the times it takes to get to the target height (going up and going down), or
• You could calculate these times using the quadratic equation and the formula:

y = 0.5*g*t^2 + vy*t

0.5*g*t^2 + vy*t - y =0

Determine the two time of flight values to hit the elevated target and the corresponding vx values. Check your answers by running the simulation with the calculated vx values. Record the data below and your selected vx values for hitting the target. Show the calculations which you used to determine the two vx values required to hit the target.

#### Hitting Target on Way Up

Hitting Target on Way Down

Initial y-Velocity = ________ m/s

Initial y-Velocity = ________ m/s

Measured y-distance = _______ m

Measured y-distance = _______ m

Measured x-distance = _______ m

Measured x-distance = _______ m

Time of Flight = _______ s

Time of Flight = _______ s

Calc'd x-velocity = _________ m/s

Calc'd x-velocity = _________ m/s

Work for Calculating x-velocity:

Work for Calculating x-velocity:

### Challenges:

1. Consider firing the same bow on the Moon, where the acceleration of gravity is -1.6 m/s/s. Can you calculate the firing angle needed to hit the target? Show your calculations in the space below. Verify your prediction by changing the acceleration of gravity and running the simulation.

Note: you need to remove the "lab walls." Scroll down the Control Panel window to access the "Lab has no walls" control. Click in this control to "remove the walls."

2. Make the target predictions for the arrow's flight on Earth using polar coordinates instead of Cartesian coordinates. Figure out how to predict the required angles of the bow to hit the balloon given a fixed arrow velocity (such as 60 m/s). Hint: Think about the use of trigonometric equations to resolve the velocity (speed and angle) into x- and y-velocity components which were used in Task 2. Explain the procedure below.

### Application Questions:

Solve the following problems using the principles which you learned in this lab.

1. A softball is hit with an initial y-velocity of 25 m/s and an initial x-velocity of 20 m/s. Determine the time of flight and the horizontal distance of the hit. PSYW

2. A football is kicked with a velocity of 30 m/s at an angle of 30 degrees above the horizontal. Determine the time of flight and the horizontal distance of the kick. PSYW

3. A horse jumps off the ground with an initial velocity of 10 m/s at an angle of 60 degrees. Determine the time of flight and the horizontal distance of the jump. PSYW