# Boy Scouts are firing model rockets in a field. One boy used a radar gun to determine his rocket had a lift-off velocity of 34 m/s. When the rocket was traveling at 17m/s, how far was it from its highest point?

##### 1 Answer

#### Answer:

It was **14.7 m** from its maximum height.

#### Explanation:

So, you know that the rockets are being launched with an initial velocity of **34 m/s**.

Since the rocket is moving straight up, it will be decelerated by the *gravitational acceleration*, **maximum height**, where it will be equal to zero.

This means that you can determine the maximum height reached by the rocket by using the formula

#underbrace(v^2)_(color(blue)(=0)) = v_0^2 - 2 * g * h#

Rearrange to solve for

#h = v_0^2/(2 * g) = (34^2"m"^color(red)(cancel(color(black)(2)))/color(red)(cancel(color(black)("s"^2))))/(2 * 9.8color(red)(cancel(color(black)("m")))/color(red)(cancel(color(black)("s"^2)))) = "59.0 m"#

Since on its way up the rocket is being decelerated by **the same velocity** as it did when it was launched.

The distance is the same, so it goes from 34 m/s to 0 m/s on its way up, and from 0 m/s to 34 m/s on its way down.

Now, after the rocket reaches maximum height, it begins to *free fall* towards the ground. Since at the top of its trajectory its velocity is equal to zero, you can use the same formula to determine the distance **17 m/s**.

#v^2 = underbrace(v_"top"^2)_(color(blue)(=0)) + 2 * g * h_"down"#

#h_"down" = v^2/(2 * g) = (17^2"m"^color(red)(cancel(color(black)(2)))/color(red)(cancel(color(black)("s"^2))))/(2 * 9.8color(red)(cancel(color(black)("m")))/color(red)(cancel(color(black)("s"^2)))) = "14.7 m"#

Therefore, the rocket is **14.7 meters** from its maximum height when it has a velocity of **17 m/s**.

**Alternatively**

You can calculate the distance from the top of the rocket's trajectory by calculating the distance it covered on *its way up* until its velocity reached 17 m/s, then subtracting this value from the max height.

#17^2 = 34^2 - 2 * g * h_"up"#

#h_"up" = (34^2 - 17^2)/(2 * 9.8) = "44.2 m"#

This means that it is

#59.0 - 44.2 = "14.8 m"#

from the top - the difference in the value comes from rounding along the way.