To determine the total distance covered by the motorboat until it comes to a complete stop after the engine is shut down, we can use the information provided about the resistance force and the relationship between velocity and distance. Let's break this down step by step.
Understanding the Forces at Play
When the engine of the boat is turned off, the only force acting against the motion of the boat is the resistance force, which is given as rV, where V is the instantaneous speed of the boat. According to Newton's second law, the net force acting on the boat can be expressed as:
F = ma
Here, m is the mass of the boat and a is its acceleration. Since the resistance force is acting in the opposite direction to the motion, we can write:
-rV = m(dV/dt)
Relating Velocity and Distance
To find the distance covered, we need to relate the change in velocity to the change in distance. We can use the chain rule to express this relationship:
F(x) = mV (dx/dV)
Substituting the expression for the force, we have:
mV (dx/dV) = -rV
Separating Variables
We can simplify this equation by dividing both sides by V (assuming V ≠ 0):
m(dx/dV) = -r
Now, we can separate the variables:
dx = -\frac{r}{m} dV
Integrating to Find Distance
Next, we need to integrate both sides. The limits for V will be from the initial velocity V0 to 0 (when the boat stops). The limits for x will be from 0 to d (the distance we want to find):
∫(0 to d) dx = -\frac{r}{m} ∫(V0 to 0) dV
Integrating the left side gives us:
d = -\frac{r}{m} [V]_{V0}^{0} = -\frac{r}{m} (0 - V0) = \frac{rV0}{m}
Final Calculation
Thus, we find that:
d = \frac{mV0}{r}
Conclusion
Therefore, the total distance covered by the boat until it stops completely is:
(A) mV0 / r
This result shows how the initial velocity and the resistance force influence the stopping distance of the boat. The greater the initial velocity or the smaller the resistance, the further the boat will travel before coming to a halt.
