# Projectile Distance as Related to Mass of an Arrow and other Factors

**Grade Level: **4th - 7th; **Type: **Physics

### Objective:

The goals of this experiment involve analyzing different aspects of projectile motions. In a first set of experiments, you evaluate how the mass of an arrow affects the distance an arrow travels. In a second set of experiments, you learn how the angle of firing affects distance. While the math that relates these variables is not introduced until high school and college physics, simply drawing graphs of the results of these experiments illustrates the existence of a relationship between these variables.

### Research Questions:

- Why does air resistance determine whether the mass of an arrow has any effect on distance?
- Does the mass of an arrow have any effect on the distance an arrow travels? Taking air resistance into account, do you think a high-mass arrowhead would penetrate a target deeper than a low-mass arrowhead?
- Does the angle from which an arrow is shot affect the distance an arrow travels? Is there a point of diminishing returns in which a greater angle minimizes distance?

In an ideal universe, such as the ones in which most physics experiments take place, mass does not matter in studies of projectile motion because where is no air resistance. However, in real life, mass exerts a large effect. The angle from which an arrow is shot also determines the arc traversed by the arrow and how far it will go. By systematic experimentation, students can take various measures of this phenomenon and graph their results.

### Materials:

If students use a computer-simulated bow and arrow, the only needed equipment is a computer, Internet access and an online computer-simulator, like the one listed in the Bibliography. Students using genuine equipment will require arrows of various weights, a regular bow and arrow, and measuring tapes. They should use a recognized archery range and have distances marked on the ground to determine how far the arrow was shot. While this equipment is available online and in specialty sport shops, students are encouraged not to spend money on this experiment if they do not have the necessary equipment.

### Experimental Procedure

**Simulated**

- Keeping velocity at 50 meters/sec, air resistance “off” and a 70-degree angle (these are your constant variables), vary the mass of the projectile by taking it to first the minimum mass and then to the maximum. When you fire the simulator, note whether there is a difference. Can you explain this result?
- Turn the “air resistance” feature on. Repeat step #1 a second time, varying the mass in 5 kg increments until mass no longer has an effect. Note how far the arrow went each time. Graph your results with mass being the y-axis and distance being the x-axis.
- Keeping velocity at 50 mg/sec, air resistance “off” and mass at 19.05 kg, experiment by shooting the arrow from a 90 degree angle. However far did the arrow go?
- Repeat step 3, varying the angle by 5 degree increments. At what point does lowering the angle become a point of diminishing returns? Repeat the experiment with air resistance on. Is there a difference?

**Real Bow and Arrow Protocol-Experiment#1**

- Identify a safe place such as an archery range to conduct this experiment. Measure the distances from the shooting range so that you can determine how far an arrow has flown.
- Weigh your arrows so that you have at least using three or four arrows of different weights. The weight of an arrow varies depending upon the materials from which the shaft is made and the weight of the arrowhead (traditionally measured in grains).
- Fire each three times, keeping the angle you hold the bow and distance you pull the bow string as constant as possible. Measure the distances and average your results together.
- Repeat Step #3 with all of your arrows.
- Graph your results. The y-axis can represent mass of the arrow and the x-axis can represent distance.

**Real Bow and Arrow Protocol-Experiment#2**

- Use the same place you used in Experiment #1. Select a weight of arrow for this experiment. Both the weight of the arrow and the distance you pull the bow-string are the constants of this experiment.
- Fire a single arrow three times, keeping the arrow parallel to the ground. Measure how far the arrow traveled. Average your results together.
- Repeat Step #2, varying the angle and shooting the arrow at least three times for each angle you select.
- Graph your results. The y-axis can represent the angle and the x-axis can represent distance.

**Terms/Concepts:** Projectile motion; Air resistance; Mass; Aangle and Distance

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