Deriving Projectile Motion Equations

[rakeru]

Homework Statement

A projectile is fired from ground level at time t=0, at an angle of θ with respect to the horizontal. It has initial speed V₀.

Part A

Find the time it takes the projectile to reach its maximum height. Express in terms of V₀, θ, and g.

Part B

Find the time at which the projectile hits the ground after having traveled through a horizontal distance, R. Express in terms of V₀, θ, and g.

Part C

Find the maximum height attained by the projectile. Express in terms of V₀, θ, and g.

Part D

Find the total distance R traveled in the x direction. Express in terms of V₀, θ, and g.

Homework Equations

V=V₀+at

y=y₀+V_y0t+0.5a_yt²

x=V_x0t

The Attempt at a Solution

For Part A, I did:

V_y=V_y0 + a_yt

0=V₀sinθ-gt

gt=V₀sinθ

t=(V₀sinθ)/g

For Part B I did:

y=y₀+V_y0+0.5a_yt²

0=0 + (V₀sinθ)t- 0.5gt²

0.5gt²=(V₀sinθ)t

gt²= (2V₀sinθ)t

t= (2V₀sinθ)/g

For Part C I did:

Vy²=Vy0²+2ay

0=(V₀sinθ)²-2gy

y= (V₀sinθ)²/2g

For Part D:

x = x₀ + V_x0t

x= V₀cosθ × (2V₀sinθ)/g

x= (V0²sin2θ)/g

Did I do this correctly?? Or did I just mess up everything.. The one I'm really confused about is Part C.. I don't know if I could use the equation that I used..

 

[haruspex]

All good.

 

[rakeru]

Really!? So I am right.. I was told by someone that part C was wrong.. o_o

 

[haruspex]

Checked it again... part C still looks right to me.

 

[Nickie]

Your solutions for Parts A, B, and D are correct. However, for Part C, you have used the wrong equation. The equation you used, Vy2=Vy02+2ay, is for finding the final velocity in the y-direction, not the maximum height. To find the maximum height, you can use the equation y=y0+Vy0t+0.5ayt2, where y0 is the initial height (in this case, y0=0) and Vy0 is the initial vertical velocity (in this case, Vy0=V0sinθ). Setting this equation equal to zero and solving for t will give you the time at which the projectile reaches its maximum height. Plugging this back into the equation for y will give you the maximum height in terms of V0, θ, and g.