Physics problem question

Question

A football kicker kicks a football with an initial speed of 22 m/s at an angle of 37 degrees above horizontal. Ignoring air resistance, how high did the ball go?

Answer 1

I completely agree with William C on this one. I just wanted to add that the latter equation that William provided requires that the projectile should be launched on level ground at least until it gets to its maximum height...i.e., the height calculated will be the vertical displacement from the starting point not the instantaneous altitude location--if that makes sense. Obviously, we can assume that a football is normally kicked on a level field.

The height equation is a quick and easy method for solving maximum height given only angle and launch speed in one step and it avoids the need for dealing with an equation that would require factoring to solve, which, as William C. indicated, would require either factoring or the calculation of final vertical velocity first and then calculating time -- as I have referred to this in my teachings as "the two step method for dodging the quadratic equation/factoring".

The y-max equation for calculating maximum height is one that you want to add to your arsenal of kinematics equations: Δy,max = v_i²(sin θ)²/[(2*(absolute value of g)]. This is the one and only equation I would recommend for the football problem. Depending on your course, you may or may not be required to memorize equations...if the latter, this is one you want to memorize in my opinion.

Answer 2

The initial velocity is v₀ = 22 m/s.

How high the ball goes depends on the vertical component of this velocity,

v_0y = v₀ sin(θ) = 22sin(37°) = 13.24 m/s

We'd like to use h = v_0yt – ½gt², but we don't know t.

There are two ways to get around this problem.

1. Since we know v_y = 0 at the maximum height, we can find t using v_y = 0 = v_0y– gt = 13.24 - 9.8t. This gives t = 13.24/9.8 = 1.35 s, which we could then plug in to h = v_0yt – ½gt² to get

h = (13.24)(1.35) – ½(9.8)(1.35)² = 8.9 m

1. Find your answer in one step using (v_y)² – (v_0y)² = –(v_0y)² = –2gh, which means

h = (v_0y)²/2g = (13.24)²/2(9.8) = 8.9 m

Either way you get the same answer.