Deriving Projectile Motion Equations from Initial Conditions

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To derive the equations for the range and total time-of-flight of a projectile launched from a height h at an angle Q, the initial conditions include the initial speed v and the effects of gravity. The range equation is incorrectly stated and should be derived from the horizontal motion, considering the time of flight. The time of flight can be calculated by analyzing the vertical motion, setting the height function to zero to solve for time. The correct time-of-flight equation incorporates both the vertical and horizontal components of the initial velocity. Understanding these relationships is crucial for accurately deriving the projectile motion equations.
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Homework Statement


Derive algebraic expressions for the range and total time-of-flight of a projectile launched with initial speed v, from a height h, and at an angle Q, above the horizontal. We were given the final equations but I am unsure of how to derive them.


Homework Equations


range=vt-.5gt^2, where g is the acceleration due to gravity and t is the time in seconds.

time= (vsinQ+((vsinQ)^2+2gh)^.5)/g


The Attempt at a Solution


Considering the ball, after it has been launched:
Fx=0
Vox=vcosQ
Fy=mg
Voy=vsinQ

?
 
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That equation for range is wrong.

The first step is to find an the height as a function of time.
Set this to zero to find the time-of-flight.
 

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