Deriving equation for minimum height in projectile motion with two give me's

[Ceeerson]

deriving equation for minimum height in projectile motion with two give me's :)

Homework Statement

A ball is launched at a 45 degree angle and lands 152.4 m away. What is the maximum height the ball will reach during its flight. I Know the answer, what I need is to find a way to derive an equation Δy = {\frac{Δxtan(ϴ)}{4}} I have played around for hours working backwards and forwards with the equations below. Its miserable.

Homework Equations

Δy = {\frac{gt}{2tan(ϴ)}}
All kinematic equations

The Attempt at a Solution

Δy = {\frac{gt}{2tan(x)}} was the best i got - with the help of the internet. Though with it i have no idea how to solve for t. Help:(

 

[voko]

What does the maximum height depend on?

What does the horizontal displacement depend on?

 

[Ceeerson]

sorry the equation in the bottom is tangent of thetaθ, I'm not sure what you are asking, they are both dependent on time, assuming gravity is constant, apparently one is able to derive an equation and calculate max height using only the three kinematic equations.but the only known variables are angle of elevation and Δx, which is the final displacement from the origin.

 

[haruspex]

what I need is to find a way to derive an equation Δy = {\frac{Δxtan(ϴ)}{4}}

Where θ is the launch angle and Δy and Δx stand for ...?

 

[Ceeerson]

oh, Δy indicates the maximum height achieved by the ball, which is achieved at half the total time since the balls starting y position is negligible, Δx indicates the total distance covered in the x position of the coordinate system, so making the ball's starting position the origin, the coordinates for the ball at max height are ((Δx/2),Δy) and the coordinates for the ball after it lands are (Δx,0)

 

[Ceeerson]

and yes, θ is launch angle above the horizontal

 

[haruspex]

Ok, so assume some launch speed u and write the equations for x and y as functions of time. Assuming level ground, what will x be when it lands?

 

[Ceeerson]

Δx = utcos(θ), Δy = ut/2sin(θ) - 1/2g(t/2)^2 Δx = 152.4m

 

[Ceeerson]

and (usinθ)/2g = Δy

 

[haruspex]

Δx = utcos(θ), Δy = ut/2sin(θ) - 1/2gt^2 Δx = 152.4m

I meant the generic equations that apply throughout the trajectory, but never mind. In the above, t is the total flight time, right? If so, the second term in the expression for Δy is wrong.
Write another equation for the height at time t.

 

[Ceeerson]

ok so I am going to start this problem over. ok so

in the y direction, Δy = (vsinθ)t - (1/2)gt^2; Vf = Vsinθ - gt; vf^2 = v^2sinθ^2 - 2gΔy

in x direction, Δx = Vcosθt

so I am guessing i rearrange for t using the x equation, so t = Δx/v(cosθ)

then i plug it into the first equation? let's see Δy = vsinθ(Δx/v(cosθ)) - (1/2)g(Δx^2/v^2(cos^2θ))

if i simplify it, Δy = (Δx)tanθ - (1/2)gΔx^2/cos^2θ

am i in the right direction ?

 

[haruspex]

in the y direction, Δy = (vsinθ)t - (1/2)gt^2; Vf = Vsinθ - gt; vf^2 = v^2sinθ^2 - 2gΔy
in x direction, Δx = Vcosθt
so I am guessing i rearrange for t using the x equation, so t = Δx/v(cosθ)

then i plug it into the first equation? let's see Δy = vsinθ(Δx/v(cosθ)) - (1/2)g(Δx^2/v^2(cos^2θ))

if i simplify it, Δy = (Δx)tanθ - (1/2)gΔx^2/cos^2θ

Those equations are all correct, but you need to think about the specific points in the trajectory at which to evaluate them. For which instants do you have data?

 

[Ceeerson]

i just have the final displacement in the x direction, i was just guessing that since the launch starts and ends on the ground that half the time wouldd be at max height, neglecting air reistance

 

[haruspex]

i just have the final displacement in the x direction, i was just guessing that since the launch starts and ends on the ground that half the time wouldd be at max height, neglecting air reistance

No, you have more than that. What is the final y displacement? What is the x displacement when at max y displacement?