How Is Maximum Height Calculated for a Rocket in Free-Fall Acceleration?

[strugglinGimp]

A rocket on the ground, accelerates straight upward from rest with constant net acceleration "a" , until time "t" , when the fuel is depleted. Here "g" is a positive number equal to the magnitude of the acceleration "a" due to gravity.

-What is the maximum height reached in terms of a, t and/or g?

i've been stuck on this for awhile so any help will be greatly appreciated

 

[Pyrrhus]

A lot of students ask this problem...

Show me what have you done so far? what is your interpretation?

 

[strugglinGimp]

i've come to see that the initial velocity is 0, the velocity of the rocket when it runs out of fuel is a*t which i got from the equation v = v_0 + at.

I'm guessing the maximum height is the distance it travels from the ground to the point where it runs out of fuel plus the distance from that point to its maximum height from free-fall acceleration.

So from the equation v^2 = V_0^2 + 2a(y) i got the distance from the ground to the point where it runs out of fuel to be a*t^2/2. This is as far as i have gotten and i am not sure how to find the other distance.

 

[Pyrrhus]

Well, the rocket's final speed when it's fuel is depleted will be the initial speed when it begins Free Fall with g acceleration and its final speed will be 0 when it reaches its max height.

 

[strugglinGimp]

so would the distance traveled to the max height during free fall acceleration be : 0= ((at^2/2)^2) +gy, and then y = -(a^2t^4)/4g?

and then would the maximum height be: (at^2)/2 - (a^2t^4)/4g ?

it doesn't look right, am i doing something wrong?

 

[strugglinGimp]

does it have anything to do with the fact that the problem stated that g is positive in this problem?

 

[Pyrrhus]

A rocket on the ground, accelerates straight upward from rest with constant net acceleration "a" , until time "t" , when the fuel is depleted. Here "g" is a positive number equal to the magnitude of the acceleration "a" due to gravity.

-What is the maximum height reached in terms of a, t and/or g?

Well the problem states g is a positive number equal to the magnitude of the acceleration a, ok. Let's make up negative and down positive then, so g is pointing the positive way and a will be pointing the negative way, also g magnitude will be equal to a.

Vector magnitudes are always positive, so I'm sure i gave the right interpretation.. if not let me know, my english is not that great.

now let's divide the problem in two parts, the one with acceleration -a and the other with acceleration g

Acceleration -a:

info:
$$V_{o} = 0$$
$$Y_{o} = 0$$

Using the equation:
$$V = V_{o} + at$$
$$V = -at$$

and:
$$Y - Y_{o} = V_{o}t + \frac{1}{2}at^2$$
$$Y = - \frac{1}{2}at^2$$

Now let's work the second part

Acceleration g
info:
$$V_{o} = -at$$
$$Y_{o} = - \frac{1}{2}at^2$$
$$V = 0$$ at max height

Using the equation
$$V^2 = V_{o}^2 + 2a(Y-Y_{o})$$
$$0 = (-at)^2 + 2g(Y_{max}+\frac{1}{2}at^2)$$

$$Y_{max} = -\frac{a^2t^2}{2g} - \frac{1}{2}at^2$$