# The model rockets listed below have just had their engines turned off when they are at the same height. All of the rockets are aimed straight up, but their speeds differ. Although they are the same...

The model rockets listed below have just had their engines turned off when they are at the same height. All of the rockets are aimed straight up, but their speeds differ. Although they are the same size and shape, the rockets carry different loads, so their masses differ. The specific mass and speed for each rocket is provided in rockets A-D.

Model Rockets

A. 30 m/s 700 g

B. 40 m/s 500 g

C. 20 m/s 600 g

D. 20 m/s 500 g

**Rank the maximum height the model rockets will reach 1- 4 with 1 being the greatest and 4 being the least. You may also choose all the same, all zero, or cannot determine. You must clearly explain your reasoning.**

Greatest---------------least

[1-------2-------3------4------all the same-----all zero------cannot determine]

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With the engines turned off, all rockets will travel with the same acceleration. This can be seen from the second's Newton's Law, that states that the net force on the object equals the object's mass times acceleration:

`F = ma`

Again, F in the equation above is the vector sum of all forces acting on the object. In the case of the rockets, the only force is the gravity. For each rocket, the gravitational force equals mg, where m is the mass of the rocket and g is the acceleration of the free fall.

Plugging mg into the equation, we get

mg = ma

There is a factor m on both sides of the equation, which can be canceled. So acceleration of each rocket will be a = g, independent on the mass.

As long as the rockets travel with the same acceleration, they obey the same equations of motions, which do not involve mass, but only velocity, time, and distance.

This would not of course be the case if the engines of the rockets were still working, because then there would be other forces involved. Also, we are assuming that the rockets are traveling in the vacuum, with no air resistance.

Let's assume the only force acting on the rockets if the force of gravity. (No other forces are specified in the problem, and without gravity, all rockets would just go on with the constant speeds, and there would be no maximum height.)

Thus, all rockets are traveling with acceleration of the free fall, g = 9.8m/s^2, directed downward. We can use the following equation of motion:

`(v_f)^2 - (v_i)^2 = -2gh` ,

where `v_f` is a final speed of the rocket after traveling distance h, and `v_i`

is the initial speed.

There is a minus on the right side of the equation because the acceleration is down, but the rockets are traveling up.

At the moment when the rocket reaches maximum height h, its speed is 0, so for all rockets `v_f = 0` .

Then it can be seen that the rocket with the highest initial speed will travel to a higher maximum height, independently of the mass. Thus,

**B> A >C = D**