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.