Motion ii
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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

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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


Motion ii

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)


Motion ii

  • az= Fz/ m

  • vz = voz + axzt

  • z = zo+ vozt+ (1/2)azt2

  • vz2 = voz2 + 2az(z – zo)


Independence of x z motion

Independence of x, , 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.


Motion ii

  • 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.


Motion ii

  • 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


Motion ii

  • 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


Motion ii

  • 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.


Motion ii

  • 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


Motion ii

  • 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


Motion ii

  • 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


Motion ii

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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