Is this Trig or Calc?

Question

I came across this trivia problem, and I'm not sure were to focus my brush-up.

I'm not looking for the answer, as that would take the fun out of it, however, I would like some tips on where to begin.

The problem is as follows:

The ballpark outfield fence is 10 feet high from foul line to foul line.

If a ball is hit 3 feet above the base plate, and the ball leaves the bat at a rate of 100mph,

what is the min angle from the horizontal required to hit a home run.

neglect effects of wind.

Now at first, I thought I could make a right triangle out of this, and make the angle go one foot higher than the back fence, but I don't think this will work.

Given that it involves a rate, I'm inclined to say it's calc.

When I think of a ballgame the ball seems to be hit at a high angle, which then arcs down.

any thoughts would be great.

Answer 1

Short answer: It's both calculus and trig.

This is a physics question. Use Newtonian mechanics to write an equation for the position of the ball. The given that the fence is 10 feet high everwhere suggest that you can work in 2 dimensions: height, and distance (from the plate). But only if the fence is uniformly the same distance from the plate. And you need to know how far the fence is from the plate, before you can solve.

Here would be a basic model for the height part of the equation:
given the hit angle Ф (from the ground), initial vertical velocity is 100 mph x sinФ. This gives a vertical velocity function of

V(t) = 100mph sinФ - 32t

The position is the integral of this, or
Y(t) = 3 feet + (100mphsinФ)t - 16t^2

always write the units, you'll have to convert mph to fps to solve!

for the distance, use

W(t) = 100mphcosФ for initial velocity, since horizontal velocity with no wind is not affected by gravity, the position

X(t) = (100mphcosФ) * t.

Once you find the distance from the plate to the fence, say D, write a function for the time to reach the fence, over Ф:
T(Ф) = D/100mphcosФ.

and for the vertical position of the ball at these times t:

Y(T(Ф)) = 3 feet + 100sinФD/100cosФ - 16(D/100cosФ)^2

finally set this equal to 10 feet and solve for Ф.

Because Y is continuous on its domain, the lowest value of Ф for which Y = 10 feet will be what you're looking for. (If there were lower Ф where Y >10, then there would also be lower Ф where Y = 10, because it's a continuous function, and you can be sure Y(0) <10.)

remember you are working with mixed units. You should probably convert mph into fps right at the start, to avoid errors.

Answer 2

Actually, I'd call it a trajectory problem. You'd need to know the distance to the foul line and the time it takes the ball to reach it at 100 mph. Then figure in the effect of gravity for that period of time (32ft/sec/sec). Add the initial 3 feet of elevation from the bat.

God Luck!

Answer 3

I got a C in algebra 2 and I'm in calc BC now getting an A. You should be fine in AB.

Answer 4

I would see this as a algebra problem. Set up your equation for the position of the ball p(t) and let your coordinate system originate at home plate. So you should have some quadratic equation that looks like an upside down parabola.

Answer 5

This is physics actually. It's a problem in projectile motion. You can solve this using Newton's laws.

Answer 6

i'd say its more geometry.... so not really either... trig involves cosine sine and tangents and any varient of that... calc is counting.... i took geometry last semester and this is similar to the questions we had...

Answer 7

I think it's a vectors question actually...

Answer 8

you'd use mechanics for this question.