# 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. How long did it take for the rocket to return to the ground? Assume the rocket went straight up and came straight down.

##### 1 Answer

It took the rocket **6.94 seconds** to return back to ground level.

#### Explanation:

So, the Boy Scouts are firing model rockets in a field.

It is important to note here that the rockets are being launched **straight up**, which means that their trajectory will be restricted to the vertical axis.

The idea here is that while it's moving straight up into the air, the rocket is being **decelerated** by the *gravitational acceleration*,

This means that the rocket will continue to go up until it reaches *maximum height*, at which its velocity is equal to **zero**.

From that point on, it will **free fall** towards the ground, this time being **accelerated** by the gravitational acceleration.

So, if its velocity is zero at maximum height, you can use its initial velocity to figure out how much time it needs to reach that height

#underbrace(v_"top")_(color(blue)(=0)) = v_0 - g * t_"up" implies t_"up" = v_0/g#

#t_"up" = (34color(red)(cancel(color(black)("m")))/color(red)(cancel(color(black)("s"))))/(9.8color(red)(cancel(color(black)("m")))/"s"^color(red)(cancel(color(black)(2)))) = "3.47 s"#

So, the rocket climbed for **3.47 seconds**. From this point on, the rocket will start its free fall motion. Since the gravitational acceleration wwill now accelerate the rocket **over the same distance**, the time ittakes the rocket to free fall will be **equal to** the time it took it to reach maximum height.

Therefore, the total time of flight will be

#t_"total" = 2 * t_"up" = 2 * 3.47 = color(green)("6.94 s")#

**Alternatively**

You can calculate the time needed for the rocket to reach ground *from maximum height* by calculating maximum height.

#underbrace(v_"top"^2)_(color(blue)(=0)) = v_0^2 - 2 * g * h implies h = v_0^2/(2 * g)#

#h = (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"#

Now you just need to calculate how much time it takes the rocket to free fall from *initial speed* for the free fall period of its flight.

#h = underbrace(v_"top")_(color(blue)(=0)) * t + 1/2 * g * t_"fall"^2#

#t_"fall" = sqrt((2 * h)/g) = sqrt( (2 * 59.0color(red)(cancel(color(black)("m"))))/(9.8color(red)(cancel(color(black)("m")))/"s"^2)) = "3.47 s"#

The **total time** it takes the rocket to return to ground level is

#t_"total" = t_"up" + t_"down"#

#t_"total" = 3.47 + 3.50 = color(green)("6.94 s")#