A problem with a projectile motion question

[keroberous]

I'm a little new here and was hoping I might be able to get some help. This is the question I'm having problems with:

A hallway is 40 m long and 3 m high. Could a professional baseball pitcher throw a ball from one end to the other before the ball hits the ground?

I haven't really even been able to get anywhere with this problem. Basically what I figured I should do was assume a height for the picture (I chose 1.5 m, but I think that choice is irrelevant). What I initially tried to do was determine the angle at which the ball would just skim the ceiling and then use that to calculate the velocity needed to ensure that the ball landed on the floor at the base of the wall 40 m away. When I tried to do that it wasn't really working, I think I may be short on known information for that to work. Then I thought that it wasn't important that the ball touch the ceiling, but I'm not really sure what approach to take. Overall, I think I'm over-complicating the problem and confusing myself further in the process.

I realize that I don't really have much accomplished in terms of my own work, at least numerically. What I'm hoping for at least is a plan of attack, and possibly a first step. I've looked around online for help in other places but anything I find is a bit over my grade level (I'm in grade 12). Thanks so much for any help in advance.

 

[gneill]

If you consider the equation for the range of a projectile that's launched from ground level, namely R = (v²/g)sin(2θ), it appears that if the angle is restricted to be less than 45° then the range is proportional to the launch angle. In other words, you want to maximize the launch angle. So it appears that your surmise that the projectile should just skim the ceiling is good.

If that's true, then given the launch height that you choose, you can fix the maximum y-component of the launch velocity. The rest should be (tedious) algebra to find the x-component of the velocity required to reach the goal.

 

[keroberous]

Ok, so here's what I ended up doing so far. I first looked at the vertical velocity, and determined what it would have to be to go 1.5 m up (i.e. to just touch the ceiling). So v₂ = 0, a = -9.8, d = 1.5, and I'm looking for v₁. the equation I used was v₁² = v₂² - 2ad and found v₁ to be 5.4 m/s [up].

I then used that to find t, using the equation d = v₁t + (1/2)at² (all values vertical again) to find t = 1.3 s.

Then I looked for the horizontal v₁, when d = 40, t = 1.3, and assuming a = 0. I used the equation d = v₁t + (1/2)at² and found that v₁ = 30.8 m/s [horizontal].

I then combined the vertical and horizontal components using Pythagorean Theorem and tan to get that v₁ = 31.2 m/s 10 degrees up from horizontal.

Any feedback on this would be appreciated. I can't help but feel this is too simple, but like I said initially, perhaps I was over-complicating things. Does this process at least make sense?

 

[gneill]

It looks good to me. 31 m/s doesn't seem like an unusual speed for a pitch.

 

[keroberous]

thanks so much for your help