Question

At launch a rocket accelerates upward at a=2g. After 12 seconds the rocket runs out of fuel and enters freefall.

a.) Determine the rocket's maximum height.

b.) How long after take off does the rocket crash to the ground?

Answer 1

At launch a rocket accelerates upward at a=2g. After 12 seconds the rocket runs out of fuel and enters freefall.

v - u =at

=> v - 0 = 2g*12 =235.2m/s final velocity when its fuel runs out just

v² - u² = 2*2g*h₁   => h₁ = 1411.2 meter height after 12 second when fuel just runs out

then it goes upward with initial speed 235.2 m/s and acceleration is downwards due to gravity

so v² - u² = 2*g*h₂   now final velocity is 0

=> 0 -(235.2)² = 2*9.8* h₂

=> h₂ = 2822.4

=> h   =  h₁ + h₂ = 1411.2 +2822.4 =4233.6m

so maximum height of rocket = 4233.6m

b) now since it falls in free fall due to acceleration g

h = 1/2gt²

=> 4233.6 = 1/2 *9.8 *t²

  => t = 29.39 seconds

this is the time to reach ground from the top

  now time to acheive it's highest point from where it's fuel runs out

and at that time initial velocity = 235.2 m/s and acceleration = g and final velocity = 0

=> v -u =gt

=>0-235.2 =9.8*t

=>t =24

so total time = time before fuel runs out + time from fuel runs out to reach it's maximum height + time to retun to ground

=12+24+29.39 = 65.39 second