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This question is challenging and involves the use of calculus. It will not appear on your final exam. We will study the effect of air resistance for...

This question is challenging and involves the ues of calculus...

a) warm up: for a freely falling object compute the velocity...

b) Now we will include air resistance. Typically are resistance takes the form Fd=-bv...

c)Wreite down Newton's second law, now including the drag force.

d) In contrast to free fall, Newton's second law now has solutions with a=0. Determine the velocity Vt, which satisfies Newton's sceond law with zero acceleration in termss of Q4.jpgb, m, and g.

e) Now follow what you did for part a). Write down Newton's second law and use the fact a=dv/dt

Q4.jpg

4. This question is challenging and involves the use of calculus. It will not appear on your final exam. We
will study the effect of air resistance for an otherwise freely falling object.
(a) Warm up: For a freely falling object compute the velocity and position as a function of time. That
is give the functions y(t) and v(t) in terms of g, the starting position yo and the starting velocity
vo. You should start by writing Newton's second law:
F = ma
and then using the fact that F, = my and a = di. Integrate Newton's second law to obtain first
v(t) and then y(t). Hint: this is constant acceleration, so you know what the answer should be.
(b) Now we will include air resistance. Typically air resistance takes the form FD = -bi, where b is a
constant which depends on the size, shape and weight of the object and the subscript D denotes
that we are talking about a drag force. Note that the faster the speed, the larger the drag force.
What are the SI units of the constant 6?
(c) Write down Newton's second law, now including the drag force.
(d) In contrast to free fall, Newton's second law now has solutions with a = 0. Determine the velocity
Ur, which satisfies Newton's second law with zero acceleration in terms of b, m, and g. This velocity
is called the terminal velocity and is the largest speed the object will reach.
Now follow what you did for part (a). Write down Newton's second law and use the fact a=
di
Integrate Newton's second law to obtain first v(t) and then y(t). Be careful with your signs. The
answer is
v(t) =UT + (vo - UT)e-m
y(t ) =yo + Urt - (UT - vo)" (e-# - 1)

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