A motor boat is traveling at a speed of 2.4 m/s shuts off its engine at t=0. How far does it travel before coming to a rest if it is noted that after 3 seconds its speed has dropped to half of its original value? Assume that the drag force of the water is proportional to v. (R=-bv)

To find the distance traveled by the motor boat before coming to a rest, we need to calculate the time it takes for the boat to come to a rest and then use this time to find the distance.

Let's start by determining the time it takes for the boat to come to a rest. We know that after 3 seconds, the boat's speed has dropped to half its original value.

Let's say the boat's original speed is v0 and its speed after 3 seconds is v3.

v3 = 0.5 * v0

Now, let's use the information that the drag force is proportional to velocity (R = -bv) to determine the time it takes for the boat to come to a rest. The drag force can be expressed as:

R = m * a

where R is the drag force, m is the mass of the boat, and a is the boat's acceleration.

Given that R = -bv, we can substitute it into the equation:

-bv = m * a

Since the boat's engine is shut off, we know that the net force acting on the boat is only the drag force. Therefore, we can also write the equation as:

-m * dv/dt = -bv

Let's rearrange the equation and separate the variables:

dv/v = b/m * dt

Integrating both sides of the equation:

∫(1/v) dv = ∫(b/m) dt

ln(v) = (b/m) * t + C

By considering the initial condition v0 at t = 0:

ln(v0) = C

Substituting the initial condition back into the equation:

ln(v) = (b/m) * t + ln(v0)

To determine the time it takes for the boat to come to a rest, we set v = 0:

ln(0) = (b/m) * t + ln(v0)

Since ln(0) is undefined, we know that the boat will never come to a complete rest. However, if we assume that "coming to a rest" means dropping to a negligible speed, we can set v to a very small positive value (close to zero), and solve for t:

ln(ε) = (b/m) * t + ln(v0)

where ε is a small positive value close to zero.

Now, we can solve for t:

(b/m) * t = -ln(v0)

t = (-ln(v0)) / (b/m)

To find the distance traveled, we can multiply the average velocity by the time.

Average velocity = (v0 + 0) / 2 = v0 / 2

Distance = Average velocity * t

Distance = (v0 / 2) * ((-ln(v0)) / (b/m))

Simplifying:

Distance = -(v0 * ln(v0)) / (2 * b/m)

Now, we can calculate the distance traveled by plugging in the given values for v0 and b/m.