Projectile Range, 1D60.40

Topic and Concept:

• Motion in One Dimension, 1D60. Projectile Motion

Location:

• Cabinet: Mechanic (ME)

• Bay: (A5)

• Shelf: #1

Abstract:

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

Important Setup Notes:

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

• This demo requires time(up to 15min) to setup and to get the targets correctly positioned on the floor

• Note that the projectile will fall or travel farther due to the height of the lecture bench, if targets are on the floor and not on the lecture bench.

• If the muzzle velocity is desired, the Pasco Interface computer and two photogates will be needed and will require additional time for setup.

Setup and Procedure:

1. Attach the Pasco launcher to the tabletop using a C-clamps.
2. Have a spotter ready to spot the impact point of the projectiles.
3. The launcher has 3 spring settings, so make sure you use the same one for all launches.
4. After loosening the proper adjustment screw, adjust the launch angle to 45° using the weighed string/protractor that is on the face of the launcher.
5. 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).
6. Shoot the projectile by pulling on the yellow string attached to the launcher.
7. Have the spotter mark the point of impact.
8. Measure the horizontal displacement, by counting the floor tiles or using a ruler.
9. Compare the measured range to the predicted to obtain the correction factor.
10. Adjust the angle now to 30° and then to 60° (chosen to be give different enough results without changing the horizontal and vertical air resistance components too much) .
11. Have one or more spotters spot the projectiles point of impact.

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!

• Projectiles will bounce and will need to be located after they are fired

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. The most convenient convention is to choose 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 then placed at the muzzle projection onto the ground.

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

|x(t)| = v_0x * t, where v_0x is the x-component of the muzzle velocity.

|y(t)| = -(g/2)* t² + v_0y * t + y₀ , where g is the acceleration due to gravity, v_0y 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 dependent upon the angle of trajectory (i.e. the angle of the barrel relative to the ground or table). The angle is usually called θ. v_0x = cos(θ)*v₀ and v_0y = 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 Muzzle height ( y = 0 ). If the projectile goes to the floor, then you need to know that height too, then (y = -h). To find the range, we first find the time at which y = 0 for t > 0. This occurs at t = (v_0y + Sqrt(v_0y² + 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.

To get the muzzle velocity, we can pull out the Pasco Interface computer and two photogates to the muzzle of the Pasco launcher. This will add a good 5 to 8 minuets to the setup time.

Videos:

• Lecture Demonstration's Youtube Channel

References:

• Projectile Motion - Wikipedia

• PASCO item page

Home