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Deriving the Range Equation: Navigating the Vertical Velocity Landscape

Uncover the path to deriving the range equation by understanding the principles of constant horizontal velocity, vertical velocity changes, and gravitational acceleration. Explore the equations and theorems involved step by step to solve for the range with a focus on there being no horizontal acceleration. Dive into the mathematics of projectile motion with this comprehensive guide to finding the range through velocity components and calculations.

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Deriving the Range Equation: Navigating the Vertical Velocity Landscape

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  1. Deriving the Range Equation Or, how to get there from here

  2. Keep in mind . . . • Horizontal velocity REMAINS CONSTANT • No net force is acting horizontally so there is no horizontal acceleration • Vertical velocity CHANGES • Acceleration due to gravity, ~9.81 m/s2 • Caused by the unbalanced force of gravity acting on the object

  3. ymax x x R = 2x

  4. ymax vi q x x R = 2x

  5. vi viy = vi sin q q vx= vicosq

  6. ymax vi viy = vi sin q q x x vx=vicosq R = 2x In the x-direction: , where t = time to top of path t

  7. ymax vi q x x R = 2x In the y-direction:

  8. ymax vi q x x R = 2x In the y-direction: At the top of the path, vfy = 0

  9. ymax vi q x x R = 2x So in the y-direction: Substituting in

  10. ymax vi q x x R = 2x So in the y-direction: Now, we have Substituting in

  11. vi Remember that the initial velocity in the y-direction =vi sin q viy =vi sin q q vx= vicosq

  12. ymax vi q x x R = 2x So we go from To

  13. The whole point here is to solve for x . . . • ( • ( • = x

  14. Remember that the range, R, = 2x = 2x = R Double-angle theorem from trig: so = R (Q.E.D.)

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