roblem 1 (Projectile motion with air resistance) We have seen in lecture that dr
ID: 3280743 • Letter: R
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roblem 1 (Projectile motion with air resistance) We have seen in lecture that dropped from rest in the preserice of bhavityand airresistanco Newton's second law of motion takes the following form: where y is the height of the object above the ground, g is the local acceleration due to gravity near the earth's surface, c is a constant, and m is the object's mass. In this assignment, you are going to find the analytical solution for the object's trajectory y(t) subject to the initial conditions y(0)=10, y(0)=0 (a) Before you try to solve this initial value problem, it will be good to do some dimensional analysis. At first glance, there appear to be six parameters involved in this IVP: y,yo, t,9, c, m. However, notice that c and m only appear as c/m. So, there are really only p 5 parameters involved: y, yo, t,g, c/m. 6) (5 points) Write down the dimensions of each of the five parameters listed in (3). (6i) (6 points) Let r be the number of independent dimensions involved in these five parameters. Find r. (ii) (15 points) According to the Buckingham Pi Theorem, there are q p-r dimen- sionless groups that can be formed from these five parameters. Choose as primary parameters the first q parameters that appear in (3). For example, if -3, choose parameters y, vo, and t. Find the q dimensionless groups associated with the primary parameters. (iv) (5 points) Now focus on the dimensionless group that contains t. Define a quantity T such that that dimensionless group reads t/T. We will see later that r has importarnt physical significance. (v) (5 points) Let I,, I12, I1I t q dimensionless groups you found. Write a general equation of the formExplanation / Answer
a) the dimensions of
y= length
y0=length
t=time
g =length time-2
c/m=length-1
b) the no independent dimensions is 3 ie y,y0,t.
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