LAB 3 - Non-Constant Acceleration
Tony Wu, Leslie Zhao, Isaiah Hernandez
September 12, 2016
Lab Goal: To learn to use Excel to do integration for kinematics.
Theory/Introduction: Solving problems involving constant acceleration is a trivially easy task. One simply needs to pick the right kinematics equations and then plug and chug. On the other hand, problems that involve non-constant accelerations are more difficult to solve as they require calculus. It is not difficult to use calculus to integrate the acceleration/velocity/position equations for simpler equations. However, when the equations become significantly more complicated, it becomes very difficult to integrate multiple times. In this lab, we learn to use Excel as a means to calculate the acceleration, velocity and positions at various times. This allows us to simply plug in the data we recorded and from the Excel spreadsheet we can solve the problem numerically.
Theory/Introduction: Solving problems involving constant acceleration is a trivially easy task. One simply needs to pick the right kinematics equations and then plug and chug. On the other hand, problems that involve non-constant accelerations are more difficult to solve as they require calculus. It is not difficult to use calculus to integrate the acceleration/velocity/position equations for simpler equations. However, when the equations become significantly more complicated, it becomes very difficult to integrate multiple times. In this lab, we learn to use Excel as a means to calculate the acceleration, velocity and positions at various times. This allows us to simply plug in the data we recorded and from the Excel spreadsheet we can solve the problem numerically.
Apparatus: This lab did not use any apparatus. We used Excel on our laptops.
Experimental Procedure: We are given the problem to solve analytically.
The Problem: A 5000-kg elephant on frictionless roller skates is going 25 m/s when it gets to the bottom of a hill and arrives on level ground. At that point a 1500-kg rocket mounted on the elephant's back generates a constant 8000 N thrust opposite the elephant's direction of motion. The mass of the rocket changes with time (due to burning the fuel at a rate of 20 kg/s) so that the m(t) = 1500 kg - 20 kg/s*t. Find how far the elephant goes before coming to rest.
We first solve the problem analytically. Newton's 2nd law gives us the acceleration of the elephant + rocket system as a function of time:
a(t)=Fnetm(t)=−8000N6500kg−20kgst=−400325−tms2
We can integrate the acceleration from 0 to t to find △v then derive an equation for v(t).In order to find out how far the elephant travels before coming to rest, we need to find the time at which the velocity of the elephant is zero. Now that we have the time, we can calculate the distance the elephant travels before stopping. We integrate the velocity from 0 to t to find △x and then derive an equation for x(t). We now plug in the time t that the elephant stops into the distance equation and get the distance that the elephant stopped. We calculated the distance that the elephant would stop is 248.7 meters.
Now we are going to use Excel to solve the problem numerically. We set up the variables that we will plug into Excel: initial mass, initial velocity, burn rate, force in Newtons, and change in time. We then have columns for time, a formula for acceleration, average acceleration, change in velocity, instantaneous velocity, average velocity, change in position and current position.
Data and Calculations: Below is the Excel spreadsheet.
Notice that in line 204, the velocity of the object reaches zero somewhere between 19.6 seconds and 19.7 seconds. We able to estimate the distance at which the elephant stops, approximately 248.693 meters.
Conclusions: Analytically, we calculated that the distance at which the elephant stops moving was 248.7 meters. Numerically, we reached a value of that is slightly greater 248.693 meters. The numbers are practically equal. In order to tell if the time interval we chose is small enough, we can pay close attention to how much each variable is changing by. In conclusion, we can see the advantages of using Excel to solve these types of problems. This method is especially useful for larger, more complicated equations that involve difficult integrations.
Given an initial mass of 5500 kg, a fuel burn rate of 40 kg/s and a thrust force of 13000 N, the elephant would travel a distance of 164 meters before coming to a stop.
The Problem: A 5000-kg elephant on frictionless roller skates is going 25 m/s when it gets to the bottom of a hill and arrives on level ground. At that point a 1500-kg rocket mounted on the elephant's back generates a constant 8000 N thrust opposite the elephant's direction of motion. The mass of the rocket changes with time (due to burning the fuel at a rate of 20 kg/s) so that the m(t) = 1500 kg - 20 kg/s*t. Find how far the elephant goes before coming to rest.
We first solve the problem analytically. Newton's 2nd law gives us the acceleration of the elephant + rocket system as a function of time:
a(t)=Fnetm(t)=−8000N6500kg−20kgst=−400325−tms2
We can integrate the acceleration from 0 to t to find △v then derive an equation for v(t).In order to find out how far the elephant travels before coming to rest, we need to find the time at which the velocity of the elephant is zero. Now that we have the time, we can calculate the distance the elephant travels before stopping. We integrate the velocity from 0 to t to find △x and then derive an equation for x(t). We now plug in the time t that the elephant stops into the distance equation and get the distance that the elephant stopped. We calculated the distance that the elephant would stop is 248.7 meters.
Now we are going to use Excel to solve the problem numerically. We set up the variables that we will plug into Excel: initial mass, initial velocity, burn rate, force in Newtons, and change in time. We then have columns for time, a formula for acceleration, average acceleration, change in velocity, instantaneous velocity, average velocity, change in position and current position.
Data and Calculations: Below is the Excel spreadsheet.
Notice that in line 204, the velocity of the object reaches zero somewhere between 19.6 seconds and 19.7 seconds. We able to estimate the distance at which the elephant stops, approximately 248.693 meters.
Conclusions: Analytically, we calculated that the distance at which the elephant stops moving was 248.7 meters. Numerically, we reached a value of that is slightly greater 248.693 meters. The numbers are practically equal. In order to tell if the time interval we chose is small enough, we can pay close attention to how much each variable is changing by. In conclusion, we can see the advantages of using Excel to solve these types of problems. This method is especially useful for larger, more complicated equations that involve difficult integrations.
Given an initial mass of 5500 kg, a fuel burn rate of 40 kg/s and a thrust force of 13000 N, the elephant would travel a distance of 164 meters before coming to a stop.
No comments:
Post a Comment