Dart Gun Lab

          If you find a dart gun (from your earlier years, perhaps) you might want to see how far the dart flies (now being a physics genius).  Using the experimental method, you can fire the dart into the air and watch how far it goes.  This is, of course, the easiest way.  But let us imagine that you are in a competition, and you must predict the distance your dart will fly if fired at a particular angle from the ground.  With the aid of a little physics, you can calculate the distance more easily and accurately that you could just by guessing.

           First you need to know how high the dart flies when fired straight up.  You will also need to know the angle at which you must eventually fire the projectile.  Projectile motion problems are generally solved after the vertical and horizontal velocities are found.  The reason for this is very simple.  Neglecting air friction, the horizontal velocity stays constant and the vertical velocity changes under the influence of gravity (9.8 m/s²).  Both of these velocities do "predictable" things: one doesn't change at all and the other changes at a fixed rate.  The horizontal velocity is used to find how far the dart will travel while it is in the air.  This constant horizontal velocity is multiplied by the time in the air to give the horizontal distance:   d_horizontal  =  V_horizontal x t    where the velocity is constant and t = time. 

 PROCEDURE:

1)  Fire your dart gun vertically (at face height) three or more times to get values for how far the dart rises before coming back down (to face height).  Record these values in the data table.  Ideally, you want to measure the rise of the center of mass of the dart.  We will attempt to measure this height using two different methods.  We want to time the dart from face height to face height with a stop watch (and then use our formulas to calculate height reached) as well as using the geometrical method involving angles and distances.

 Method one (stop watch):  time from ground to ground:

trial 1: ____________s

trial 2: ____________s

trial 3: ____________s         Average: _______s        Fall time: _______________s

 Method two (geometrical)

distance from observer to gun: _______________ 

trial 1: angle noted with protractor: _______°

trial 2: angle noted with protractor: _______°

trial 3: angle noted with protractor: _______°   average angle: ___________º

 

 Method one:  average height: __________m

 Method two:  average height: __________m

 Average height of the dart:     _________m

2)  Calculate the velocity of the dart leaving the gun by using the height to which the dart rose (average from step 1) and setting the initial kinetic energy equal to the potential energy at the highest point.

 K.E.  =  P.E.           1/2 m v²  =  m g h    (solve for v)         velocity_initial = _____________m/s

 3) To find out if the gun is firing the dart consistently (at the initial velocity just obtained), place the gun, lying flat, on the edge of the lab counter.  Measure the height of the dart above the floor (center of dart to floor) and calculate the fall time (t = √ 2d/g):  __________s. 

Calculate how far the dart should fly horizontally in that time (d = V_initial • t):  ___________m.  

Measure this distance along the floor from directly beneath the end of the dart gun out in the direction it will fire.  Place your target there.  Fire the gun 3 times to determine how close it comes to the target distance.

Test distance 1: ___________ , test distance 2: ___________, test distance 3: ______________

average distance: __________m     

percent difference between calculated distance and average actual distance: _____________
 (hint: if this distance is too large in your opinion recalculate the initial velocity and then repeat this step.)

 4) Using a chosen angle of 45° from the horizontal, fire your dart three more times from the ground.  Fire at the same angle each time.  Make sure the end of the barrel is as close to the ground as possible while maintaining the correct angle (inverting the gun may help). Measure the horizontal displacement from where the dart is fired to where it lands each time.  Record these values.

trial 1: ____________m     trial 2: __________m    trial 3: _________m     average: __________m

   5) Find horizontal and vertical components of the initial velocity:

           V_vertical  à  sin 45°  =     X
                                                         V_initial                                    V_vertical  =  __________ m/s

   

         V_horizontal  à  cos 45°   =  _Y
                                               V_initial                            V_horizontal  =  __________ m/s

   

6)  Find the time in flight:

           t_up  =  V_vertical           time_up = _________s
                                g

 

         t_total =  2 • t_up        time_totoal =  _____________s

7)  Find distance traveled:

           d_horizontal  =  V_horizontal   •  t_total            distance_horizontal  =  ______________m

 8) percent difference between your answer (# 7) and the actual distance (# 4 above):

  _________________%

 9)  Using the above data/formula calculate the distance your dart gun would fire its projectile if using

 an angle of __________°.  Write in pen your estimated distance prior to test flight.

 Estimated distance gun will fire dart: _____________m   

 10)  Fire your dart at the target apparatus.  Measure the distance it actually travels over three trials. 

 trial 1: ______m   trial 2: ________m     trial 3: _________m   average distance:___________m

    Percent error between calculated distance (# 9) and actual distance (# 10): ____________%

 Questions and Problems: (use formulas from formula sheet, lab, and the following)

Maximum Range of Dart Gun:    V² sin(2θ)/2g        (V =  V_initial)    This is a maximum when θ = 45°.
Height of Projectile (shot at angle other than 90º):  (V_initial • sin θ)(t) - ½gt² 
Time in flight:  (V_initial • sin θ) / g

 1.  Given that the acceleration due to gravity equals 9.8 m/s², and that a dart reaches a height of 5.0 meters when fired vertically upward, at what initial velocity must it leave the gun?

   2.  If the dart in problem #1 were to be fired at the same initial velocity, but at an angle of only 35° from horizontal, what would be the horizontal and vertical components of the initial velocity?

  3.  How long would it take the dart in the previous problem to reach the highest point in its flight if fired vertically?    If fired at 55° from horizontal?

 4.  What would be the total time in the air for the dart in the previous problems if it were fired vertically?  If fired 55° from horizontal?

  5.  How far would the dart in the previous problems fly horizontally before it hit the ground if fired at 35° from horizontal?

  6.  A big dart gun shoots a dart 20.0 meters vertically when fired straight up into the air.  Find the initial velocity of this dart and how far it will fly if it were to fired at a 40° angle to the horizontal surface with the same initial velocity as when fired vertically.