March 8: Lab #3 Non-Constant Acceleration

Lab #3

Non-Constant Acceleration

Ricardo Gonzalez

Josue Luna

3-8-17

Our goal is to find how far an object will travel with a non-constant acceleration in the opposite direction of the initial motion. 

1. Introduction 

So far many of our experiments have assumed acceleration is constant, however in real life cases, that is not always the case. When lets say a rocket is launched from NASA with a constant thrust, the mass of the rocket changes with time as the rocket fuel burns. In this case, the force of thrust is unchanged while the mass decreases therefore acceleration is much larger. 

2. Procedure

What we have is an elephant moving down an inclined hill toward a flat horizontal surface. At the moment the elephant is moving across the horizontal surface, the elephant has an initial velocity of 25 m/s. Notable, the elephants mass is 5,000 kg and a rocket of 1500 kg is strapped onto the elephant with the direction of thrust opposite to the motion of the elephant. As fuel is burned from the rocket, the rockets mass begins to decrease given by a function. We want to calculate how long must the hill be in order for the elephants velocity to approach zero and begins to move in the opposite direction.

3. Data

Initial velocity of the elephant = 25 m/s

mass of elephant = 5000 kg

mass of the rocket = 1500 kg

Constant Rocket thrust = 8000 N

the function m(t)= 1500 kg - 20 kg/s*t

Newtons first law Fnet = ma

4. Results

5. Analysis
We have the means to find the value of x, our maximum distance that the elephant will travel with the given information in two ways. Both analytically and numerically. Analytically, we can use Newton's second law to find our function for acceleration by setting a in terms of t (time). By taking the integral of our acceleration function with limits (0,t) , we are able to find our change in velocity. What we have is a function in terms of( time with an initial velocity constant. Once again, we can take the integral of our velocity function [v(t)] with limits (0,t) to give us our position function in terms of time. When we set our velocity function equal to zero, we are able to find how long the elephant took to come to rest.The time we calculated was 19.69 seconds. Lastly, we are able to substitute our value of time into our position function and find our total distance the elephant traveled so that its velocity was 0. We can see that the value of x was calculated as 248.7 meters.
From a numerical approach, we are able to set an excel file in such a way so that we are able to find the precise value of x as we set our time interval to be a much smaller value. As shown in the photo above we have set: the first column as the time interval, the second column as the acceleration, the third column as the average acceleration, the forth column as the change in velocity, the fifth column as the velocity, the sixth column as the average velocity, the seven column as the change in distance, and finally our eighth column as the distance. When we set our time interval of one second, we can see that at the 19th second, the position changed from a positive value to a negative value. This tells us that between the 19th second and the 20th second,the elephant changed direction. As we changed our time interval to a tenth of a second, we now can see that between the 19.7 and 19.8 second, our direction of motion changed. While we did not use a smaller time interval, we would see that the value of x would begin to change in such a small amount of distance relative to the total distance traveled that any further time interval changes are needed.
If we were to change the mass of the elephant to 5500 kg, the fuel burn rate to 40 kg/s*t, and the thrust force to 15000 N, we can see that our values change. our new value of x is roughly 12 meters between 12.7 and 12.8 seconds.

6. Conclusion
When accelerations are constant, finding positions and velocities at given times are much simpler. When an acceleration is not constant, we must find a function for the acceleration and find its components of velocity and position by taking a few integrals. Although useful, at times the functions and or integrals may be much more complicated that anticipated. When such a case arises, we can use numerical approaches to find the same results with out the headache.