1. (20 points) An experimentalist finds that air resistance causes a force actin
ID: 1326679 • Letter: 1
Question
1. (20 points) An experimentalist finds that air resistance causes a force acting opposite to the motion on an object that is proportional to the velocity squared. F = ky? (a) Let y(t) be the position of a falling object at time t. Using second law of Newton. show that the differential equation for the position is given by where g is the acceleration due to gravity and k is a positive constant that depends on properties of the object. (b) The terminal velocity of an object is the y that a falling object can have. Once an object reaches it's terminal velocity, the velocity no longer changes This happens when the happens when the opposing resistive force is equal to the gravitational force Recall that the equation of motion without resistive forces yields a velocity that ithout bound. Calculate the terminal velocity in terms of k,m linearly increases without bound. Calculate the terninal velocity in terms of k, m and g. (e) Verify that a solution to the differential equation is given by y(t) = yo log(1+1e-2V with initial position yo Determine the limiting behavior as t oo. Interpret this result. In particular how does it relate to the terminal velocity.Explanation / Answer
according to newtons second law
F = m a
a = F / m
where m is the mass and a is the acceleration
to get the postion we use the equation of motion
S = ut + (1 / 2) a t2
as the motion is in vertical direction
S = y (t)
y (t) = u t + (1 / 2) (F / m) t2
but F = k v2
y (t) = u t + (1 / 2) (kv2 / m) t2
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