"Don't let Biomechanics Students Play with Guns"
Biomechanics Lab
Projectiles with Dart Guns

Julieanne Abendroth-Smith
University of Utah

Learning Objective
 
 

To increase understanding of the basic principles and equations of projectiles.
  Equipment
 
 
 
dart gun with soft darts

reflectors on dart and gun as noted

ruler

protractor

 tape measure

stop watch

 

Background
 
 

You are trapped behind the sofa, and you need to tag your sibling hiding behind the overstuffed chair. Your darts have all fallen short of the mark. What do you do to increase the distance the dart will fly?
 
 

Soft-tipped darts are a good example of projectiles. Different factors such as height of release, angle of release and even speed of release are easily manipulated to affect the trajectory, or path the dart will follow. The equations of projectiles can be used to determine how fast the dart leaves the gun, and how high and far a dart may fly.

Projectiles can be examined through the horizontal and vertical components of movement. The following equations describe that movement:
 
 

Vertical Direction (y) Horizontal Direction (x)

dy = vy1t + 1/2 gt2 dx = vx1t

vy2 = vy1 + gt vx2 = vx1

ay = g = - 9.81 m/s/s or -32 ft/s/s ax = 0

vy22 = vy12 + 2gdy
 
 

 where d = displacement

v = velocity

a = acceleration

g= acceleration due to gravity
 
 
 
 

Activities and Questions ** Record your events both manually, and if available, a video or auto digitizing system.
 
 

(Mark the gun as well as the dart with reflectors. The location of the markers allows for the angle of release to be determined by the digitizing unit. Experiment with other marker locations, such as on the floor, to better determine when the dart contacts the floor.)
 
 

1. Separating the vertical and horizontal components of projectiles
 
 

Stand with a dart loaded in the gun. Keep the gun level to the horizontal, so no vertical velocity is imparted to the dart. Drop a second dart simultaneously with firing the gun. How do the landing times of both darts compare?
  2. Angle, height, and speed of release
 
  Place the butt of the dart gun on the ground. Fire the gun several times, changing the angle of release. What happens to the paths (trajectories), distances, (range), and heights the dart reaches? Does the time in the air change? What angle allows the dart to fly the further? Which angle gives the dart the most time in the air?

Manually measure some angles with a protractor. Fire the gun again. Pick the gun up one meter off of the ground, but keep the same angle, and shoot. What changes as a result of firing the gun at a greater height of release?

Keep the angle and height of release consistent. Most dart guns have a set speed of release. What can you do to change the speed of release, without changing how the gun operates? As an example, try moving your arm forward as the you pull the trigger. Does adding some initial velocity of your own change anything about the dart's flight?
 
 

 3. Determining the speed of release and components at different angles The resultant initial velocity a dart leaves a dart gun is the speed of release. This is accomplished by determining the initial horizontal and vertical components of velocity, while using different angles of release. The height the dart travels at apex of flight can also be determined for each angle.
  Record the following values for each dart's flight.
 
 
 
Angle of Dart
Height of barrel from floor (dy)
Time in air (s)
Range traveled (dx)
60 degrees
      
45 degrees
      
30 degrees
      

 
 
 
 
 

Calculate the following values for each dart's flight.
 
 
 
Initial vertical velocity
Initial horizontal Velocity
Resultant velocity
Height at apex
       
       
       

 

Comparisons of Results
 
 

Generate the graphs of the x and y components of displacement, velocity and acceleration from the video or auto digitizing unit, as well as the resultant velocity. Locate the same information for each angle as measured or calculated above in part 3.
 
 
 
Angle of release Height of barrel from floor (dy) Time in air (s) Range traveled (dx) Initial Vertical velocity Initial Horizontal Velocity Resultant velocity Height at apex
               
               
               

 

How do your results compare with the results produced by the video system? Where are the largest sources of error?