Differences between revisions 1 and 2
⇤ ← Revision 1 as of 2012-07-20 23:32:31 →
Size: 3936
Editor: srnarf
Comment:
← Revision 2 as of 2012-07-20 23:42:46 → ⇥
Size: 4837
Editor: srnarf
Comment:
Deletions are marked like this. Additions are marked like this.
Line 24: Line 24:
||Target || Provide One|| || ||Target  or Tape|| Provide Own|| ||
Line 28: Line 28:
 * '''''It is a good idea to use a clamp to keep the launcher in one place through the demonstration.'''''  * '''''It is a good idea to use a C-clamp to keep the launcher in one place through the demonstration.'''''
Line 36: Line 36:
 1.  1. Use the black loading rod, push one of the Pasco projectiles down the barrel to the desired spring setting ( a click will be heard for each setting reached - third click setting is the most energetic).
 1. Shoot the projectile by pulling on the yellow string attached to the launcher.
 1. Have the spotter mark the point of impact.
 1. Measure the horizontal displacement.
 1. Compare the measured range to the predicted to obtain the correction factor.
 1. Adjust the angle now to either 30° or 60° (chosen to be give different enough results without changing the horizontal and vertical air resistance components too much) .
 1. Using the corrected range equation, predict the range at the new angle and mark the spot.
 1. Have one or more spotters spot the projectiles point of impact.
 1. Using the corrected range equation,
Line 40: Line 48:
 * ''''' DO NOT point the launcher into the audience!'''''
Line 48: Line 57:
|'''x(t)'''| = v,,0,x,, * t, where v,,0,x,, is the x-component of the muzzle velocity
|'''y(t)'''| = -(g/2)* t^2^ + v,,0,y,, * t + y,,0,, , where g is the acceleration due to gravity, v,,0,y,, is the y-component of the muzzle velocity, and y,,0,, is the height of the muzzle off the ground.		 |'''x(t)'''| = v,,0x,, * t, where v,,0x,, is the x-component of the muzzle velocity
|'''y(t)'''| = -(g/2)* t^2^ + v,,0y,, * t + y,,0,, , where g is the acceleration due to gravity, v,,0y,, is the y-component of the muzzle velocity, and y,,0,, is the height of the muzzle off the ground.
Line 51: Line 60:
The amount of muzzle velocity in each component is purely dependant upon the angle of trajectory (i.e. the angle of the barrel relative to the ground). The angle is usually called θ. v,,0,x,, = cos(θ)*v,,0,, and v,,0,y,, = sin(θ)*v,,0,, where v,,0,, is the total muzzle velocity. The amount of muzzle velocity in each component is purely dependant upon the angle of trajectory (i.e. the angle of the barrel relative to the ground). The angle is usually called θ. v,,0x,, = cos(θ)*v,,0,, and v,,0y,, = sin(θ)*v,,0,, where v,,0,, is the total muzzle velocity.
Line 53: Line 62:
In this demo, we are concerned only with the range which is our total displacement in the x-direction after the projectile comes back to the ground ( y = 0 ). To find the range, we first find the time at which y = 0 for t > 0. This occurs at t = (v,,0,y,, + Sqrt(v,,0,y,,^2^ + 2*g*y,,0,,))/ g In this demo, we are concerned only with the range which is our total displacement in the x-direction after the projectile comes back to the ground ( y = 0 ). To find the range, we first find the time at which y = 0 for t > 0. This occurs at t = (v,,0y,, + Sqrt(v,,0y,,^2^ + 2*g*y,,0,,))/ g

[:PiraScheme#Mechanics: Table of Mechanics Demonstration]

[:MEEquipmentList: List of Mechanics Equipment & Supplies]

[:Demonstrations:Lecture Demonstrations]

Projectile range, 1D60.40

Topic and Concept:

  • Motion in One Dimension, [:MotionIn1D#ProjectileMotion: 1D60. Projectile Motion]

Location:

attachment: mainPhoto

Abstract:

Shoot a projectile out of the Pasco laucnher at 45° then calculate the range for 30° or 60°, and place the target accordingly.

Equipment

Location

ID Number

Pasco Launcher

ME, Bay B1, Shelf #2

Pasco Projectiles

ME, Bay B1, Shelf #2

Target or Tape

Provide Own

Important Setup Notes:

  • It is a good idea to use a C-clamp to keep the launcher in one place through the demonstration.

Setup and Procedure:

  1. Attach the Pasco launcher to the tabletop using a C-clamp.
  2. Have a spotter ready to spot the impact point of the projectile.
  3. After loosening the proper adjustment screw, adjust the launch angle to 45° using the weighed string/protractor already on the launcher.
  4. Use the black loading rod, push one of the Pasco projectiles down the barrel to the desired spring setting ( a click will be heard for each setting reached - third click setting is the most energetic).
  5. Shoot the projectile by pulling on the yellow string attached to the launcher.
  6. Have the spotter mark the point of impact.
  7. Measure the horizontal displacement.
  8. Compare the measured range to the predicted to obtain the correction factor.
  9. Adjust the angle now to either 30° or 60° (chosen to be give different enough results without changing the horizontal and vertical air resistance components too much) .
  10. Using the corrected range equation, predict the range at the new angle and mark the spot.
  11. Have one or more spotters spot the projectiles point of impact.
  12. Using the corrected range equation,

Cautions, Warnings, or Safety Concerns:

  • DO NOT look down the barrel of the Pasco launcher - a spontaneous launch could take your eye!

  • DO NOT point the launcher into the audience!

Discussion:

The equation of projectile motion for constant acceleration can be derived by starting with the acceleration and integrating twice with respect to time to find the position. It is most convenient to use a coordinate system where the horizontal motion is purely in the x-direction, and the vertical motion is purely in the y-direction. The origin is placed at the muzzle projection onto the ground.

r(t) = x(t)+y(t)

|x(t)| = v0x * t, where v0x is the x-component of the muzzle velocity |y(t)| = -(g/2)* t2 + v0y * t + y0 , where g is the acceleration due to gravity, v0y is the y-component of the muzzle velocity, and y0 is the height of the muzzle off the ground.

The amount of muzzle velocity in each component is purely dependant upon the angle of trajectory (i.e. the angle of the barrel relative to the ground). The angle is usually called θ. v0x = cos(θ)*v0 and v0y = sin(θ)*v0 where v0 is the total muzzle velocity.

In this demo, we are concerned only with the range which is our total displacement in the x-direction after the projectile comes back to the ground ( y = 0 ). To find the range, we first find the time at which y = 0 for t > 0. This occurs at t = (v0y + Sqrt(v0y2 + 2*g*y0))/ g

After plugging in this time into the x-position equation we find the range R = (v0*cos(θ)/g) * (v0*sin(θ)+Sqrt(v02*cos2(θ)+2*g*y0)) which if maximized gives a theta of 45°.

This is of course neglecting air resistance which is speed-dependent. This is why a shot is taken first then a prediction is made. Since the muzzle velocity is roughly constant (at least for the same spring setting and ignoring random error), comparison to the "air-resistanceless" equation will give an empirical correction factor. This should allow for a fairly close trajectory prediction. It will be a little off since the air resistance does have a θ-dependance.

attachment: photo

attachment: photo

attachment: photo

attachment: photo

Videos:

References:

[:Instructional:Home]

fw: Projectile_Range (last edited 2013-07-12 18:18:20 by localhost)