04-March-2015 Lab3: Non-constant acceleration problems/ activity solutions

Lab 3: Unconstant acceleration

We can calculate every problem by using Newton's law if the acceleration is constant, but what if the acceleration is not constant.

Today we are going to do is to figure out how we get acceleration, velocity, time, and position if the acceleration is not constant.

1. We are doing it analytically 

EX: A elephant is 5000kg and has a constant velocity 25 m/s at the beginning, and the elephant has a rocket on his back. Suddenly, the rocket starts to fire and gives 8000N to decrease the elephant.

For this question, we know that it has initial velocity v0 = 25 m/s, and then it has deceleration  because the rocket. So we are using the F= ma  Newton's second law and integrate acceleration with respect to time t to set up the all the formula.

Here is picture and calculation.

After we done our calculation, we got a(t), v(t), x(t).
To find how far the elephant goes before coming to rest. We should find when velocity is zero.

So  v = (25 - 400*ln(325) + 400ln(325-t) = 0  We got t = 19.7s
when plug the x (19.7)  we got  x = 248.7 m
We want to know if we use the newton's law when the acceleration is not constant, it's still work or not. 

2. We calculate the it by using numerical integration.

For it's unconstant acceleration, we are going to do is to make a every small time interval to make in that small interval velocity vs time graph it's a straight line which gives a constant acceleration, then we can use the newton's law.

For example. show the picture.

We can see that acceleration in this picture is not constant, so if we want to calculate we should give a small enough interval so that in that line will be like straight, and then we can apply then Newton's law.

Therefore, we let that interval be △t = 0.1s, and then we use excel to input date. By using this formula, we get our data.

When △t = 0.1s,

t = t + △t,

a = F/m =-400/(325-t)

△a = (a0 + a1) /2 (a0 = -1.23m/s^2)

△v = △a *  △t

v = v0 + △v (v0 = 25m/s)

v-avg = (v0 + v)/2

△x = v-avg * △t

x = x0t + △x (x0 = 0)


We are using this formula to get the acceleration, velocity, and distance


After we plug t into a we can get all the data from excel
Here is the picture of data.

As time t is increasing, all the data such as a, a_avg .. are changing

Here is all the date from t = 0 to t =20.1s

Now we are going to test our data by comparing the function we get and see if it's correct or not.

Now we are using the equation we got before with a(t), v(t), x(t).  

When we change the time we got different a, v ,and x such as when t some number t1, it will give a(t1), v(t1), and x(t1)

For example;
we plug when t= 10.2s

it gives  a(10.2)=-1.27./s^2
v(10.2)= 12.24m/s

x (10.2)=190.3m

we using this results and compare with the data from excel when t = 10.2s, we find  that in excel when t =10.2 s, a=-1.27m/s^2, v= 12.24m/s, x=190.3m

Here are some others samples.

When t = 12.1s

So we can say when t = 19.7 s, we got the distance x from excel x = 248.7m .
We can see that the results we calculated by using the formula is same exactly the answer in excel.

Conclusion

  We can compare first results and second results, it's turns out that they are almost exactly same. Also, we should let the interval small enough, that's mean that it changes very small, then the interval you chose is fine.