# Math Help - A Rocket in the Air

1. ## A Rocket in the Air

A rocket loaded with fireworks is to be shot vertically upward from the ground level with an initial velocity of 200 feet per second. When the rocket reaches a height of 400 feet on its upward trip, the fireworks will be detonated. How many seconds after liftoff will this take place?

2. Hello, sologuitar!

A rocket loaded with fireworks is to be shot vertically upward from the ground level
with an initial velocity of 200 feet per second.
When the rocket reaches a height of 400 feet on its upward trip, the fireworks will be detonated.
How many seconds after liftoff will this take place?
We are expected to be familiar with this formula: . $h \:=\:200t - 16t^2$
. . where $h$ is the height of the rocket (in feet) $t$ seconds after liftoff.

When is $h \,=\,400$ ?

We have: . $200t - 16t^2 \:=\:400 \quad\Rightarrow\quad 2t^2 - 25t + 50 \:=\:0$

. . which factors: . $(2t-5)(t-10) \:=\:0$

. . and has roots: . $t \:=\:\tfrac{5}{2},\:10$

The fireworks will be detonated $2\tfrac{1}{2}$ seconds after liftoff.

3. ## After...

Originally Posted by Soroban
Hello, sologuitar!

We are expected to be familiar with this formula: . $h \:=\:200t - 16t^2$
. . where $h$ is the height of the rocket (in feet) $t$ seconds after liftoff.

When is $h \,=\,400$ ?

We have: . $200t - 16t^2 \:=\:400 \quad\Rightarrow\quad 2t^2 - 25t + 50 \:=\:0$

. . which factors: . $(2t-5)(t-10) \:=\:0$

. . and has roots: . $t \:=\:\tfrac{5}{2},\:10$

The fireworks will be detonated $2\tfrac{1}{2}$ seconds after liftoff.
After providing the formula, the question becomes crystal clear.

4. I have a serious problem with this question! If an object is thrown upward, perhaps from a catapult, the only force acting on it is gravity and that formula applies. But the whole point of a rocket is that the rocket engine continues to fire so there is an additional upward force on it. If this really were a rocket we would have to know the thrust from the rocket engine, which is not given, to answer the question.

5. ## Yes but...

Originally Posted by HallsofIvy
I have a serious problem with this question! If an object is thrown upward, perhaps from a catapult, the only force acting on it is gravity and that formula applies. But the whole point of a rocket is that the rocket engine continues to fire so there is an additional upward force on it. If this really were a rocket we would have to know the thrust from the rocket engine, which is not given, to answer the question.
Yes, but remember that a detailed rocket math question is found or taught in physics. This question comes from my precalculus textbook. The author is just interested in knowing if students can munipulate equations.

6. It's still a bad problem. The student is expected to assume things that are contrary to the information given.