A body weighing 64 Ib is dropped from a height of 100 ft with an initial velocity of 10 ft/sec. Assume that the air resistance is proportional to the velocity of the body. If the limiting velocity is known to be 128 ft/sec, find (a) an expression for the velocity of the body at any time t and (b)

Question

A body weighing 64 Ib is dropped from a height of 100 ft with an initial velocity of 10 ft/sec. Assume that the air resistance is proportional to the velocity of the body. If the limiting velocity is known to be 128 ft/sec, find (a) an expression for the velocity of the body at any time t and (b) an expression for the position of the body at any time t.

Accepted Answer

(a) Locate the coordinate system as in Fig. 7-5. Here w = 64 Ib. Since w = mg, it follows that mg = 64, or m = 2 slugs. Given that [latex]v_{1}[/latex] = 128 ft/sec, it follows from (7.6) that 128 = 64/k, or [latex]k=\frac{1}{2}[/latex] Substituting these values into (6.4), we obtain the linear differential equation

[latex]\frac{d v}{d t}+\frac{1}{4} v=32[/latex]

which has the solution

[latex]v=c e^{-t / 4}+128[/latex] (1)

At t = 0, we are given that v = 10. Substituting these values into (1), we have [latex]10=c e^{0}+128, ext { or } c=-118[/latex] . The velocity at any time t is given by

[latex]v=-118 e^{-t / 4}+128[/latex] (2)

(b) Since v = dx/dt, where x is displacement, (2) can be rewritten as

[latex]\frac{d x}{d t}=-118 e^{-t / 4}+128[/latex]

This last equation, in differential form, is separable; its solution is

[latex]x=472 e^{-t / 4}+128 t+c_{1}[/latex] (3)

At t = 0, we have x = 0 (see Fig. 7-5). Thus, (3) gives

[latex]0=472 e^{0}+(128)(0)+c_{1} \quad ext { or } \quad c_{1}=-472[/latex]

The displacement at any time t is then given by

[latex]x=472 e^{-t / 4}+128 t-472[/latex]

_____________________________________________________________________________________________________________________________

Eq. (7.6) : [latex]v_{l}=\frac{m g}{k}[/latex]

Eq. (6.4) : [latex]y^{\prime}+p(x) y=q(x) y^{n}[/latex]

Question 7.13: A body weighing 64 Ib is dropped from a height of 100 ft wit...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions