Question

Question #43d72

Answer 1

62.8 meters

Explanation:

If `x` is the height at any time t, then `x'' = (d^2x)/(dt^2)`

And at earth's surface this is a constant 9.81 meters per second per second. (rounding to 3 significant digits because that's what we did on our slide rules oh-so-many years ago.)

then `v(t) = dx/dt = -9.81t + c`

We're saying -9.81t because we're assigning negative values to the downward direction, positive values to the upward direction.

We can calculate the value of constant c because we know the value of v at time 0: 12m/s.

so we now know that `v(t) = 12 - 9.8t = dx/dt`

We know the ball will travel upwards while decelerating, eventually reaching velocity 0 at the apex of its travel. We can solve for the time when this occurs:

`12 - 9.8t = 0`
`12 = 9.8t`
`t = 12/9.8 = 1.22` seconds (rounding to 3 digits again)

We know that at time t = 2 * 1.22, or 2.44 seconds, the ball is back to it's original height, and is travelling downwards at 12 m/s. (We're ignoring wind resistance).

Since we're told that the ball's total flight time is 5 seconds, we know that it has `5 - 2.44 = 2.56` seconds flight time remaining.

At this point, we can start a completely new derivation. We have known initial conditions v(0) = -12 m/s, and total elapsed time will be 2.56 seconds.

So we have velocity `v(t) = -12 - 9.8t`

...and we can integrate to find x(t):

`x(t) = -12t - (9.8/2)t^2 + c`

...and we know that at time t = 2.56 the ball's height will be zero.

With this, we can solve for constant c, which will be the tower's height (and the ball's height at time t = 0).

`x(2.56) = 0 = (-12 * 2.56) - (4.9 * 2.56^2) + c`

`-30.7 - 32.1 + c = 0`

c = 62.8 meters. (rounded to 3 significant digits in all calculations)

...and you can see that we sidestepped the need to calculate the max height that the ball reached.

Answer 2

`sf(63color(white)(x)m)`

Explanation:

Setting the ball at the origin and using the convention "up is positive" we can use:

`sf(s=ut+1/2at^2)`

This becomes:

`sf(s=(12xx5)-(1/2xx9.81xx5^2))`

`sf(s=60-122.625=-62.625color(white)(x)m)`

This means the height of the tower can be rounded to 63 m.