You are walking around your neighborhood and you see a child on top of a roof of a building kick a soccer ball. The soccer ball is kicked at 33° from the edge of the building with an initial velocity of 21 m/s and lands 71 meters away from the wall. How tall is the building that the child is standing on?

Question

whpalmer4 · Answer

Okay, you need to decompose that 21 m/s @ 33 degrees into horizontal and vertical components.  Do you know how to do that?

anonymous · Answer

21cos(33) ?

anonymous · Answer

and 21sin(33)?

whpalmer4 · Answer

Yeah.  Which is which? :-)

anonymous · Answer

21cos(33) is horizontal 
31sin(33) is vertical

anonymous · Answer

answer. hor. is 17.612
vet. 11.437

whpalmer4 · Answer

Okay, so we have two problems to solve: 

1) what is the time of flight
2) how high is the building

whpalmer4 · Answer

we can write an equation for the height of the ball as a function of time:

$$h(t) = -9.8t^2 + v_0t + h_0$$where $-9.8$ is a constant (in units of m/s^2) for the acceleration due to gravity, $v_0$ is the initial vertical velocity (computed above), and $h_0$ is the initial height (read: the height of the building)

however, we only have 1 equation and two unknowns, so that isn't enough to find both $t$ and $h_0$.

whpalmer4 · Answer

However, we also know that the ball travels 71 m horizontally on its trip to the ground.  $$d = r*t$$$$71 m = r*t$$where $r$ is the horizontal component of the initial velocity.

With that, we can find the value of $t$.  With the value of $t$, we can solve for $h(t) = 0$ which will give us $h_0$, the height of the building.  Give it a shot and I'll check your answer.

anonymous · Answer

t=4.03
ho=113.069

whpalmer4 · Answer

Yes, that's what I get as well.

anonymous · Answer

so the answer is 113.069?

whpalmer4 · Answer

Yes, unless we both made the same mistake...

whpalmer4 · Answer

Here's a plot of the height of the ball vs time (blue), the distance the ball has traveled horizontally vs. time (purple) and x = 71 meters (olive)

Note that the point at which the diagonal line (the horizontal distance traveled by the ball) intersections 71 meters is exactly over the point where the height of the ball crosses the x-axis (hit the ground).  That's what we want to see!