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Motion II. 2 and 3 dimensional motion. Components of Motion . Motion in 2 dimensions X component Y component Motion in 3 dimensions X component Y component Z component. Motion in x direction is independent of motion in y direction and z direction.

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motion ii

Motion II

2 and 3 dimensional motion

components of motion
Components of Motion
  • Motion in 2 dimensions
    • X component
    • Y component
  • Motion in 3 dimensions
    • X component
    • Y component
    • Z component
slide3

Motion in x direction is independent of motion in y direction and z direction.

Separate set of equations of motion for each direction.

equations of motion
Equations of Motion
  • ax= Fx/ m
  • vx = vox + axt
  • x = xo+ voxt + (1/2)axt2
  • vx2 = vox2 + 2ax(x – xo)
  • ay= Fy/ m
  • vy = voy + ayt
  • y = yo+ voyt + (1/2)ayt2
  • vy2 = voy2 + 2ay(y – yo)
slide5

az= Fz/ m

  • vz = voz + axzt
  • z = zo+ vozt+ (1/2)azt2
  • vz2 = voz2 + 2az(z – zo)
independence of x y z motion
Independence of x, y, z motion
  • Motion in the x direction is independent of motion in the y or z directions.
  • Motion in the y direction is independent of motion in the x or z directions
  • Motion in the z direction is independent of motion in the x or y directions.
nerf gun experiment
Nerf Gun Experiment
  • In class, a nerf gun was fired horizontally from a height of 3’10” and struck the ground at a distance of 16’10”.
  • Calculate the muzzle velocity of the projectile.
  • Calculate the time of flight of the projectile.
slide8

Neglecting aerodynamic drag, the projectile leaves the muzzle with a velocity vo = vox.

  • The projectile as it leaves the muzzle has no velocity in the y-direction, i.e. voy = 0.
  • The only force on the projectile after it leaves the muzzle is the force of gravity.
  • The acceleration in the y direction (up and down) is g = 32.2 ft/s2.
slide9

Knowing the initial y component of velocity is 0, the acceleration in the y direction is 32.2 ft/sec2, and the distance to the floor is 3’10”,

  • 3’10” = 3.833 ft = y – yo
  • ay = 32.2
  • y = yo+ voyt +0.5ayt2
  • 3.833 = 0.5 x 32.2 x t2
  • t = 0.488 sec
slide10

In that time of 0.488 sec, the projectile travels

  • 16.833 ft in the horizontal direction.
  • vx = 16.833/0.488 = 34.49 ft/sec
  • So the muzzle velocity is 34.49 ft/sec
elevated nerf gun
Elevated Nerf Gun
  • Consider the same nerf gun, but now elevated at an angle of ϴ⁰ to the horizontal.
  • The muzzle velocity is vo
  • The horizontal velocity vox = vocosϴ
  • The vertical velocity is voy= vosinϴ
  • vyvo
  • vx
slide12

The only force acting on the projectile after it leaves the muzzle is gravity – in the y-direction.

  • The projectile will arc up, stop rising, and arc down to hit the ground.
  • We can then calculate how high the projectile will rise and the time it takes to reach that maximum height.
slide13

vy2 = voy2 + 2ay(y – yo)

  • voy = vosinϴ
  • ay = - g
  • vy2 = vo2sin2ϴ + (2)(-32.2)(y – yo)
  • If ϴ = 30⁰ and vo = 28.0 ft/sec
  • 0 = (28.0)2 (0.5)2 – 64.4 (y – yo)
  • (y – yo) = 3.04 ft
slide14

To calculate the time for the velocity in the y-direction to go from (14) ft/sec to 0,

  • vy = voy + ayt
  • 0 = 14 – (32.2)t
  • t = 0.435 sec
  • The projectile then begins to fall and it takes another 0.435 sec for it to hit the ground. A total time of flight of (2)(0.435) = 0.87 sec
slide15

During that entire 0.87 sec, the projectile is moving in the x-direction at its initial speed.

  • There is no force in the x-direction causing it to speed up or slow down.
  • Its speed in the x- direction is (28)(0.866) ft/sec = 24.25 ft6/sec
  • In 0.87 sec, the projectile travels (0.87)(24.25)
  • = 21.1 feet in the x-direction before it impacts the ground!
review
Review
  • Motion in 2 dimensions
    • X component
    • Y component
  • Motion in 3 dimensions
    • X component
    • Y component
    • Z component
slide17

Motion in x direction is independent of motion in y direction and z direction.

Separate set of equations of motion for each direction.

class activity
Class Activity
  • Consider a rifle with a muzzle velocity of 3,000 ft/sec firing at ϴ⁰ to the horizontal.
  • Calculate the range and time to impact as a function of ϴ.
  • Create an excel worksheet and plot range vsϴ.
  • At what value of ϴ would you get the maximum range? Analytically and graphically!
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