Mass of our rocket: 543 grams Initial Thrust: 131 newtons
Take care to notice that the Initial thrust is stated. As the rocket flies, the thrust will be constantly changing. We will get to that soon. For now we will be concerned only with calculating what can be called instantaneous thrust. The word instantaneous means the thrust at any one instant in time.
Acceleration means a change in velocity. The equation for acceleration is generally written as:
Over the years we humans have derived a method of answering that question: How much mass? Mass is the defined as the resistance to acceleration. (Deceleration is the same as acceleration, just in the other direction.) How do we do that?
We take that same equation and reformat it to: m = F / a or mass = Force divided by acceleration.
Question: But that's the same equation for Force. Are we using something to define itself?
After practicing a few times, I use the force meter and press on the cube with exactly one newton of force for exactly one second. Then I stop pushing and you measure the velocity of my cube and find that it is moving at 1 m/sec. The mass was accelerated for one second and the velocity was then equal to 1 m/sec.
Science has decided that the mass of that object is one kilogram.
If I put two of the cubes together and press with the same force for the same time, you will measure a velocity of 1/2 m/sec. If I take the first cube, cut it in half, then the result will be a velocity of 2 m/sec.
We had to start somewhere, so with that one experiment we demonstrated the definitions of mass, newton, acceleration, and one second.
There is more in that simple equation, F = am, than is at first apparent.
Back to the question: Why divide by m? Because if I increase the mass, the acceleration is reduced and the velocity is reduced. That means mass must be in the denominator.
Now we have the numbers for the the amount of thrust, the F part of the equation. We also have the numbers for the mass of our rocket, the m part of the equation. Plug the values into the equation. But, remember the section above: Keeping The Terms. Putting it all together we have:
Read that out loud. Something is missing. Acceleration is measured as meters/sec2. We don't have any meters or sec2.
The amount of force needed to accelerate one kilogram at the rate of one meter per second squared.
That is the definition of a newton. Any place we have newton it can be replaced with: 1 * kilogram * meter / sec2.
Let us look at one newton written as a fraction.
1 * kilogram * meter ____________________ = 1 newton sec2
(The "1" is usually presumed and not written but I want to be clear.)
Now let us take our equation: a = F/m and replace our symbols F and m with the values we have already determined. From the previous lesson our force was 132.72 newtons. Our mass was 542.65 grams. We convert the grams to kilograms to get 0.54265 kilograms. When written as a fraction we have:
132.72 kilogram * meter _________________________ = acceleration sec2 * 0.543 kilograms
Make sure you see the newtons and the mass. The mass is easy, the newtons is a bit more difficult.
Do you see a factor of 1 in that equation. If not, look for kilograms divided by kilograms. Lets remove the kilograms and we have:
132.72 meter _________________________ = acceleration sec2 * 0.543
Is this starting to look familiar? Let us do the division and clean up the denominator some.
132.72 / 0.543 = 240.73 so we have:
240.73 meter ________________ = acceleration sec2
We have distance divided by time squared.
Remember our discussion with the car. That is an acceleration. If our rocket were to accelerate that fast, after one second it would have a velocity velocity of more than 240 meters per second. At that rate the rocket would need less that one second to get to the other end of two and a half football fields. That is moving pretty fast. If a car accelerated that fast it would go from zero to more than 500 miles per hour. In one second.
We multiplied the area of the piston by the length of the stroke. The area of piston corresponds to
the area of the nozzle. The stroke of the piston in the engine corresponds to the length of the
water column left by the rocket. In the engine we had the stroke and multiplied by the
area of the top of the piston to get the volume.
In this problem, we have the area (of the nozzle) and we already have the volume of the water. We want to calculate the length of that cylinder, the length of the column of water. So divide the area into the volume of water. The volume was 0.5 liters or 500 cm3 and the nozzle area is 3.8 cm2.
Remember all the discussion about terms. Write down the equation, do the division, and see if you get a length in centimeters. Hint: we will be dividing cubic centimeters by squared centimeters. Said another way the division problem will contain these terms:
500 cm * cm * cm ________________ 3.8 cm * cmDo the division and remove the factors of one and we have: 131.6 cm or 1.316 meters.
Yes it does. Rather surprising isn't it.
Because as the water squirts out the back, the rocket has less mass. That means it should accelerate faster. But, at the same time, the air expands to take up the space the water vacated. That means that the pressure is reduced and there is less force on the water. Simultaneously, the air expands very quickly causing the temperature to go down. That further reduces the pressure.
The result is that the mass is being reduced, but the force (the air pressure) Thai produces the thrust is also being reduced.
Think about it. From these few lessons you have sufficient information to answer the question. But it is a difficult question to answer. You will not answer it for certain until you put some numbers together. And they have to be put together just right. We will start putting those facts together and come up with our answer in the next lesson.
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