What is the formula for θ as a function of time in a projectile's path?

[k-rod AP 2010]

Homework Statement

A ball is thrown horizontally from the top of a cliff with an initial V_o(at t=os). At any moment , its direction of motion makes an angle θ to the horizontal.

1)Derive a formula for θ as a function of time, t, as the ball follows a projectile's path.

Homework Equations

The Attempt at a Solution

components: V_x=V_oxcosθ V_y=V_oysinθ

then i plugged the x component to find t:V_x=x/t... t=x/(V_oxcosθ)

next i plugged t into a y position function Y=Y_o+V_oyt+1/2at^{^2}

0=h+(V_oysinθ )(x/V_oxcosθ)+(1/2*-9.8*(x/V_oxcosθ)^{^2})

this be correct because if u were given all of the initial heights and velocities the only variable left to solve for would be θ, giving you your answer right?

 

[PhanthomJay]

That's a pretty long equation. The horizontal velocity is always constant: v_x = v_ox.
The vertical velocity is just v_y = gt.
You can solve for theta as a function of t using basic trig properties of a right triangle.

 

[k-rod AP 2010]

hey thanks, but u can only solve for theta using trig for a 1-d motion problem, but in 2-d the object follows a parabolic path so its velocity must be broken into components.

the t=x/v_o cos(theta) gives you time which you plug into the y position function, which accounts for the parabolic path.

if anybody else knows what to do, any help would be aprreciated

 

[k-rod AP 2010]

Does anybody else know how to solve this, I am really curious what the correct procedure is...

 

[Condensate]

If it helps you, I would start by sketching out a graph showing the function of time (Y) and its relationship to t. You can completely ignore X since it will have nothing to do with how quickly Y changes. Y is only a function of time.

From there on out, follow PhanthomJay's suggestions.

 

[Condensate]

After re-reading your OP, I'm sorry for replying with an incorrect response. I misunderstood your question (whoops!)

...its direction of motion makes an angle θ to the horizontal.

Use the equations for your X and Y components of your velocity and work out θ using trig.

 

[k-rod AP 2010]

no worries Condensate =P.. and ok i will try that when it comes around to AP review time, thanks for your help.