Projectile range, 1D60.40

Topic and Concept:

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

Location:

• Cabinet: [:MechanicsCabinet:Mechanic (ME)]

• Bay: [:MechanicsCabinetBayA1:(A1)]

• Shelf: #1,2,3..

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.

Important Setup Notes:

• It is a good idea to use a 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.

Cautions, Warnings, or Safety Concerns:

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

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)| = v_0,x * t, where v_0,x is the x-component of the muzzle velocity |y(t)| = -(g/2)* t² + v_0,y * t + y₀ , where g is the acceleration due to gravity, v_0,y 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. v_0,x = cos(θ)*v₀ and v_0,y = 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 = (v_0,y + Sqrt(v_0,y² + 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.

Videos:

• [https://www.youtube.com/user/LectureDemostrations/videos?view=1 Lecture Demonstration's Youtube Channel]

References:

• [https://en.wikipedia.org/wiki/Projectile_motion Projectile Motion - Wikipedia]

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