Technical Mathematics Worksheet



Catapult Worksheet

Projectile Motion

1. Consider throwing a ball into the air (with some angle, not straight up), what would the path of the ball be? Sketch your idea on the following graph, label where the ball hits its maximum height and maximum distance from the starting location:

Students should graph a parabola or parabolic like shape:

Discuss the different shape graphs and ask students to justify their graphs with physical explanations.

Extensions:

• Should the graph start at the origin?

• What factors might change the shape of the graph? (Prompt them to think about size/mass of object, air resistance, gravity, initial speed)

For the following questions we will need to use a few formulas about projectile motion. An object is considered a projectile behaving under projectile motion rules when it starts with an initial speed and initial angle and has only the force of gravity and air resistance working upon it to bring it back to the ground.

Definition of variables:

[pic]initial speed of object at time of launch

[pic]= gravity

[pic]catapult launch angle

The following three variable t, h, and d are instantaneous values at time t:

[pic] time elapsed from launch

[pic] height of ball at time t

[pic]= horizontal distance of ball at time t

The following three variables T, H, D are total or maximum values:

[pic] total time of flight (from launch to landing)

[pic]maximum height achieved by ball

[pic] total horizontal distance achieved by ball (at landing)

Projectile Motion Formulas:

To find launch angle [pic]:

[pic]

To find initial speed [pic]of object:

[pic]

2. A soccer ball is kicked into the air, it reaches a maximum height of 3 meters and lands on the ground 12 meters from where it was kicked:

a. Use the formula given above and your scientific calculator to determine the launch angle [pic] Show all your work mathematically in the space provided and then put your answer in the box provided.

Using the formula given above:[pic]

[pic]

| |

|[pic] |

b. Use the launch angle [pic], the formula above and your scientific calculator to find the initial speed of your ball as it leaves the catapult, [pic]. Show all your work mathematically in the space provided and then put your answer in the box provided.

Using the formula given above: [pic]

[pic]

| |

|[pic] |

3. A ball is launched into the air from a cannon, it reaches a maximum height of 120 feet and lands on the ground 1525 feet from where it was launched:

Note: The units in this problem are given in US Customary Units (in feet) rather than in International Standard (SI) Units (metric). You could add a quick lesson on conversions factors at this step.

1525 feet ( 464.82 m

120 feet ( 36.576 m

a. Use the formula given above and your scientific calculator to determine the launch angle [pic] Show all your work mathematically in the space provided and then put your answer in the box provided.

Using the formula given above:[pic]

[pic]

| |

|[pic] |

b. Use the launch angle [pic], the formula above and your scientific calculator to find the initial speed of your ball as it leaves the catapult, [pic]. Show all your work mathematically in the space provided and then put your answer in the box provided.

Using the formula given above: [pic]

[pic]

| |

|[pic] |

If you would like to convert into SI units (or did from the beginning, the result would be:

[pic]

Place and securely hold the catapult on a level surface and launch the ball, note where it lands. Take the height measuring stick and place it at the halfway point between the launch and landing. (You will need to measure the maximum height achieved for the next several launches.)

4. Holding the catapult steady (do not make any changes to the settings), launch the ball several times and record the maximum height, maximum distance and time of flight. Once you have collected data from 3-6 launches, average the data at the bottom of each column.

The table is filled in with data gathered in a specific experiment. Students’ data should look similar when they measure the calculations. The data collected was measure in US customary units (feet) but the equivalent SI units (meters) are provided for the averages.

|Launch |Maximum Height |Maximum Distance |Time of Flight |

|1 |11.5’ |93’ |1.83 |

|2 |13’ |90’ |1.70 |

|3 |12.5’ |85’ |1.90 |

|4 |12’ |87’ |1.65 |

|5 |12.5’ |91’ |1.73 |

|6 |13.5’ |89’ |1.60 |

| | | | |

|Average |[pic] |[pic] |[pic] |

| |[pic] |[pic] | |

5. Use the formula given above and your scientific calculator to determine the launch angle [pic] Show all your work mathematically in the space provided and then put your answer in the box provided.

Using the formula given above:[pic]

[pic]

| |

|[pic] |

6. Use the launch angle [pic], the formula above and your scientific calculator to find the initial speed of your ball as it leaves the catapult, [pic]. Show all your work mathematically in the space provided and then put your answer in the box provided.

Using the formula given above: [pic]

[pic]

| |

|[pic] |

Knowing the launch angle and the initial speed of a projectile object, we can make a graph of the path of the projectile through the air using the following formulas to determine the x and y coordinates at different times along the path:

[pic]horizontal coordinate of the ordered pair on the graph

[pic]vertical coordinate of the ordered pair on the graph

[pic] and [pic]

Sometimes a little rearrangement of the formulas makes the computation easier:

[pic] and [pic]

For the next problem we are going to use these formulas along with the values we found for the launch angle [pic] and the initial velocity [pic]:

[pic] and [pic]

7. Divide your total time from the last column of collected data above by 10. This will be your increment of time. Fill in the table below in increments starting at 0 and increasing by the increment determined.

To find the increment of time: [pic]

|Elapsed Time |x – Coordinate |y-coordinate |(x, y) |

|t | | | |

| |x |y | |

|0 |0 |0 |[pic] |

|.173 s |8.72 |4.41 |[pic] |

|.346 s |17.4 |7.87 |[pic] |

|.519 s |26.2 |10.37 |[pic] |

|.692 s |34.9 |11.91 |[pic] |

|.865 s |43.6 |12.46 |[pic] |

|1.038 s |52.3 |12.16 |[pic] |

|1.211 s |61.0 |10.8 |[pic] |

|1.384 s |69.8 |8.55 |[pic] |

|1.557 s |78.6 |5.2 |[pic] |

|1.73 s |87.2 |1 |[pic] |

8. Plot your ordered pairs [pic] on the graph below; adjust the scale on your axes as necessary.

Below is an Excel generated graph of the points found in the above table. Students should have a parabolic shaped graph after they plot the points in their table.

9. Connect the points you graphed. Was your earlier graphical prediction about projectile motion correct? What type of a curve is this?

10. In order to find the launch angle of the catapult arm that will maximize the distance covered by the ball, you will need to manipulate the formula [pic] and solve for D. Show your work mathematically and then place the formula for D in the box given.

Start with [pic]

Square both sides to eliminate the square root:

[pic]

Now multiply both sides by [pic]:

[pic]

Divide both sides by g to isolate D:

[pic]

| |

|[pic] |

11. Use the initial launch velocity [pic] found in #7, the formula you found for D in #11, and your scientific calculator to find the horizontal distance D the ball will travel for the following launch angles of the catapult arm.

Using the formula: [pic]

[pic]

|Launch Angle |Distance |

|[pic] |D |

|[pic] |18.1 ft |

|[pic] |52.2 ft |

|[pic] |80 ft |

|[pic] |98.1 ft |

|[pic] |104.4 ft |

|[pic] |98.1 ft |

|[pic] |80 ft |

|[pic] |52.2 ft |

|[pic] |18.1 ft |

12. Based off your calculations in the table above, what would be the best launch angle [pic] for the catapult arm if you want to get the ball to travel the maximum possible distance from the catapult?

At [pic]the distance is maximized at 104.4 ft.

13. If you set the catapult arm at this angle [pic], how far will the ball land from the catapult?

At [pic]the distance is maximized at 104.4 ft.

14. Now you need to adjust the catapult base so the launch angle will give you a maximum distance. You will need to change the launch angle from the initial launch angle found in #6 to the angle [pic]you determined would net the maximum distance for the ball, found in #13. Determine the adjustment factor between your two angles and place your answer in the box. For example, if you found the initial launch angle in #6 was [pic]and the angle that would net the maximum distance in #13 to be [pic] then your catapult will need to be adjusted by [pic].

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|Adjustment Factor [pic] |

The test catapult had a launch angle of [pic].

Sind the angle that maximizes the distance is [pic], the adjustment factor is found:

[pic]

15. In order to launch the ball at the angle that will achieve the maximum distance, you will need to create an angle with the floor and the catapult base by placing wooden blocks under the front of the catapult. Carefully measure the angle between the catapult and the floor until you have raised the base up to the adjustment factor found in the last step. Hold the catapult securely at the correct angle. Fire the catapult and record the distance. This should be longer than for an unadjusted launch. Record this distance.

| |

|Distance = 104 ft |

After students have adjusted their catapults, they should find the distance of the launch to be close to 104 ft.

16. Suppose you want to launch the ball from the catapult to hit a target that lies at a point set at 2/3 of the maximum distance found above. Use the formula given below and your scientific calculator to determine to determine the angle required to achieve this distance. Show all your work mathematically in the space provided and then put your answer in the box provided.

[pic]

[pic] of the maximum distance of 104 ft is:

[pic]

Using the formula [pic]:

[pic]

| |

|[pic] |

17. Set the catapult to launch at the angle determined for this distance. The catapult base will again have to be adjusted to achieve this requirement. Place wooden blocks appropriately until you have adjusted the launch angle to the proper degree measurement and hold securely at the correct angle. Fire the catapult and record the distance. The distance should be close to desired.

After students have adjusted their catapults,

they should find the distance of the launch to be close to 70 ft.

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