We have no equation to work from so let’s make one up.

Let v be the vertical speed given to the ball when it was kicked. The acceleration of gravity is about 32 ft per sec per sec (rate of change of speed). Let h be the height of the balcony and t be the time measured from when the ball is kicked. We call the function H(t) that gives the height above the ground of the ball at any time t. Initially H(t)=H(0)=h because the ball is at the height of the balcony.

t seconds after the ball has been kicked it rises at speed v and would reach a height of vt feet. But gravity acts on the ball in the opposite direction. This causes it to lose speed. For each second that passes it loses 32 ft per sec, so in t seconds it loses 32t feet per second. The average speed lost over this time is the average of 0 (right at the start) and 32t, so that’s half of 32t=16t. The distance it loses is 16t×t=16t², because distance (height)=speed times time. So we need to subtract this from the speed it gained by kicking and we get vt-16t². The ball was kicked initially from a height of h (the height of the balcony above the ground). So now we can write the equation of the function: H(t)=h+vt-16t². Remember that H is the distance above the ground. We can review this equation. When t=0 H=h, the height of the balcony. That’s OK. When t=1, H=h+v-16. The ball will be above the balcony if v-16 is positive. That means v must be greater than 16 ft/sec. However, eventually gravity wins as time goes on. When H=0, the ball reaches the ground so we have a quadratic equation: h+vt-16t²=0 which needs to be solved. The equation can also be written: 16t²-vt-h=0. This will give two solutions for t, one of which will be negative and is discarded because it refers to a time before the ball was kicked. So we take the positive solution as the time it takes for the ball to reach the ground.

To find out the height of the balcony we will need to know v and t.

EXAMPLE

If the ball takes 4 seconds to reach the ground and it is kicked with a vertical speed of 60 ft/sec, then:

h+60×4-16×16=0; h+240-256=0; h-16=0; h=16 feet.

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