uppose we wish to model the height of a projectile shot directly up at time zero at time t. Let x(t) be this function. If we assume that the only forces acting on the projectile after time zero are gravity and air resistance, one way to model the resulting motion would be the ODE m = -mg – kv, where k is a constant of proportionality determined by the shape of the projectile and v(t) represents the velocity of the function at time t.
uppose we wish to model the height of a projectile shot directly up at time zero at time t. Let x(t) be this function. If we assume that the only forces acting on the projectile after time zero are gravity and air resistance, one way to model the resulting motion would be the ODE m = -mg – kv, where k is a constant of proportionality determined by the shape of the projectile and v(t) represents the velocity of the function at time t.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:Suppose we wish to model the height of a projectile shot directly up at time zero at time t. Let x(t)
be this function. If we assume that the only forces acting on the projectile after time zero are gravity
and air resistance, one way to model the resulting motion would be the ODE m = -mg – kv,
where k is a constant of proportionality determined by the shape of the projectile and v(t) represents
the velocity of the function at time t.
(a) Solve the ODE for v(t) assuming that the initial velocity is the number vo.
(b) Determine what happens to v(t) as t → .
(c) This limiting value identified in part b is often called the terminal velocity. Suppose there are
two objects with masses m1 and m2 with m1 > m2. Using part a, which object will have the
higher terminal velocity? Explain.
(d) Suppose we used our old model m de
a limit to how fast an object in this model will move (i.e. under gravitational acceleration
without air resistance)? Explain. What about the model in part a? Explain.
-mg which did not account for air resistance. Is there
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In part (a) we have to solve the linear differential equation for v(t)
and then inn part b) we have to find the limiting solution for v(t)
and finally in part c) we need to compare the terminal velocity for two objects of different masses.
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