High School

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A 96 lb weight is dropped from rest in a medium that exerts a resistive force with magnitude proportional to the speed. Find its velocity as a function of time if its terminal velocity is -128 ft/s.

A) \( v(t) = 32e^{-t/4} - 128 \)

B) \( v(t) = 32(1 - e^{-t/4}) - 128 \)

C) \( v(t) = 32(1 - e^{t/4}) - 128 \)

D) \( v(t) = 32e^{t/4} - 128 \)

Answer :

Final answer:

The correct equation for velocity as a function of time for a weight dropped in a medium with a resistive force proportional to speed and with a terminal velocity of -128 ft/s is option B) v(t) = 32(1 - e^-(t/4)) - 128. This equation captures the velocity starting from rest and asymptotically approaching the terminal velocity over time.

Explanation:

To find the velocity as a function of time for a 96 lb weight dropped from rest in a medium with a resistive force proportional to the speed, we can use the knowledge that the terminal velocity is reached when the gravitational force is equal to the resistive force. At terminal velocity, the acceleration is zero and the object falls at a constant speed. The equation of motion for such a scenario typically has the form of:

v(t) = Vt(1 - e-(b/m)t),

where Vt is the terminal velocity, b is a constant proportional to the magnitude of the resistive force, and m is the mass of the object. Using the given terminal velocity of -128 ft/s (negative because the direction of motion is downward), our equation simplifies to

v(t) = -128(1 - e-(b/m)t).

To solve for b/m, we could set v(t) equal to half of the terminal velocity, but since we are not provided with such information, we usually look at the options available:

  • A) v(t) = 32e-(t/4) - 128
  • B) v(t) = 32(1 - e-(t/4)) - 128
  • C) v(t) = 32(1 - e(t/4)) - 128
  • D) v(t) = 32e(t/4) - 128

Options that represent the physics of the situation should asymptotically approach the terminal velocity as time increases and should start at v(0) = 0 since the object was dropped from rest. Option A) does not start at zero and option D) increases beyond the terminal velocity which is not possible. Option C) has the wrong sign in the exponential, which would lead to an increase in velocity instead of an asymptotic approach to the terminal velocity. Therefore, option B) is the correct choice as it represents a velocity starting from rest and approaching -128 ft/s as time goes on. Thus, the correct velocity as a function of time is:

B) v(t) = 32(1 - e-(t/4)) - 128.

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