# If a stone is dropped at an altitude of 174.9 m from a helicopter which is ascending with a velocity of 20.68 m/s, how long does the stone take to reach the ground?

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The problem I am having is that I am confused about using **positive** #g# because gravity is directed towards the Earth and the stone is falling in that direction or if I should be using **negative** #g# because the stone is falling opposite to the direction in which the helicopter is ascending, upwards??

It would really help if you guys could explain why g should be negative or positive. Thanks guys. :))

The problem I am having is that I am confused about using **positive** **negative**

It would really help if you guys could explain why g should be negative or positive. Thanks guys. :))

##### 2 Answers

8.45 seconds.

#### Explanation:

The direction of 'g' when talking about acceleration depends on the coordinate system we define. For example if you were to define downwards as the positive 'y' then g would be positive. Convention is to take upwards as positive so g will be negative. This is what we shall use, also we take the ground as

We can look at this much more closely by starting from scratch with Newton's second law. When the stone is dropped it has an initial velocity but the only force acting on it is due to gravity. We have defined upwards as the positive y direction so by Newton's second law we can write

This is because the stone will accelerate towards the earth, which we have defined as the negative direction.

Integrating this expression gives:

This models the velocity and makes sense if you think about it. When it's released, it will have the same velocity as the helicopter and will thus move upwards for a time but as time progresses it will stop and then begin to fall.

To find displacement, we integrate again:

Apply initial condition

To solve for the time to reach the ground, set

This is definitely a job for the quadratic formula:

Taking

We discard the negative solution so therefore the stone takes 8.45 seconds to hit the ground.

We know that

As I said earlier, with an upwards coordinate system

Set

Now use

so

This means that the stone stops momentarily at

Now we don't have any pesky initial velocities to contend with, just a straight fall from this height:

As upwards is positive, falling will result in a negative displacement so

8.45s

#### Explanation:

The helicopter is asceding with a velocity

*Considering the point of dropping the stone from helicopter as origin we proceed as follows*

If **upward** initial velocity be taken **positive** then **downward acceleration (g)** should be taken as **negative** and **downard displacement (h)** should also be considered **negative**.

*Now calculation of time (t) of reaching ground*

So we have

Inserting these in equation of motion under gravity **(comprising the variables h,u,g,t)** we get

**The same equation(1) will be obtained if we reverse the direction**