Projectile motion while playing catch

[Daniel Luo]

Homework Statement

You are playing catch with a friend in the hallway of your dormitory. The distance from the floor to the ceiling is D, and you throw the ball with an initial speed v₀=√(6gD). What is the maximum horizontal distance (in terms of D) that the ball can travel without bouncing? (Assume the ball is launched from the floor).

Homework Equations

Range = [v₀²*sin(2θ)] / g

y = y₀ + xtan(θ)-1/2gx²/(v₀²*cos²(θ))

The Attempt at a Solution

I tried to use the equations and solve for x, without much luck. All my calculations ended up with a lot of unknows...

 

[SteamKing]

You are given v0, y0 and you know that y <= D. g is a constant. What other unknowns did you come up with? It's very hard to check your work if you don't supply it for review.

P.S.: What is x supposed to be? Shouldn't you have a variable for time somewhere?

 

[haruspex]

P.S.: What is x supposed to be? Shouldn't you have a variable for time somewhere?

Looks like Daniel has substituted for t using t = x / v₀ cos(θ). But it has been done incorrectly in the v₀ sin(θ) t term, resulting in v₀ tan(θ) instead of x tan(θ).

Daniel, think about the fact that the ball must just avoid hitting the ceiling. What does that tell you about v₀ sin(θ)?

 

[Daniel Luo]

#Haruspex

Thanks for the correction. It is corrected now.

 

[Daniel Luo]

Ok so I tried this:

The max. height is:

D = [(6gD)*sin(θ)] / (2g)

which simplifies to:

D = 3Dsin(θ)

Hence: sin(θ) = 1/3.

I tried using this for the horizontal distance:

R = [(6gD) * (2 cos(θ) sin(θ))] / g

I found cos(θ) by: √(1²-(1/3)²) = (√8)/3

Next,

R = [ 6gD * 2 * (√8)/3 * 1/3*] / g = [(8√2)/3]D

But this answer is incorrect according to the answers which says R = (4√2)D.

Can you see what I've done wrong?

 

[CAF123]

Ok so I tried this:

The max. height is:

D = [(6gD)*sin(θ)] / (2g)

Your method is good, but check this equation.

 

[Daniel Luo]

#CAF123

OF COURSE! It is the square of the y-component of the initial velocity! So it's sin squared theta. Thanks for pointing it out and letting me think my self :-).

 

[Hoang]

Can you tell me how did you come up to this please ??

R = [(6gD) * (2 cos(θ) sin(θ))] / g

 

[haruspex]

Can you tell me how did you come up to this please ??

It's quoted in post #1 as a standard equation (but in the sin(2θ) form. It is the range of a projectile at angle θ fired from ground level.
It is not hard to derive it from first principles. Just write the two usual horizontal and vertical distance at time t equations, set the vertical distance to zero and eliminate t. Discard the x=0 solution.

 

[Touuka]

The max. height is:

D = [(6gD)*sin(θ)] / (2g)

Is this the y- equation? And if it is, are we allowed to eliminate the vy0?

 

[haruspex]

Is this the y- equation? And if it is, are we allowed to eliminate the vy0?

As noted in post #7, it is wrong.

 

[Touuka]

As noted in post #7, it is wrong.

Thank you!

 

[Adesh]

I can see no restriction that we cannot throw the ball totally horizontally. Does that game of catch implies/restricts the totally horizontal throw?

 

[jbriggs444]

I can see no restriction that we cannot throw the ball totally horizontally. Does that game of catch implies/restricts the totally horizontal throw?

Nice. "Without bouncing" allows rolling.