A power boat of mass 460 kg is cruising at a constant speed of 8.9 m/s. The propeller provides a drive force of 765 N. The driver of the boat shuts off the engine, and the boat coasts to a halt. Assume-contrary to fact-that the resistive force due to the water is constant, independent of the boat's speed.

(a) How far does the boat coast?

(b) How much time does it take for the boat to come to rest after the engine is turned off?

To determine the distance the boat coasts, we need to calculate the time it takes for the boat to come to a stop after the engine is turned off. Once we have the time, we can use the formula for distance traveled during constant acceleration to find the distance.

Let's start by finding the acceleration of the boat after the engine is turned off. We can use Newton's second law of motion:

Force = mass × acceleration

The only force acting on the boat after the engine is turned off is the resistive force due to the water. We can equate this force to the product of mass and acceleration:

Resistive force = mass × acceleration

The resistive force is constant and equal to 765 N, so we have:

765 N = 460 kg × acceleration

Solving for acceleration:

acceleration = 765 N / 460 kg

Now that we have the acceleration, we can determine the time it takes for the boat to come to a stop. We can use the formula for velocity change during constant acceleration:

Final velocity = Initial velocity + (acceleration × time)

The final velocity is 0 m/s because the boat comes to a stop. The initial velocity is 8.9 m/s, and the acceleration is the value we calculated earlier. We can rearrange the equation to solve for time:

Time = (Final velocity - Initial velocity) / acceleration

Time = (0 m/s - 8.9 m/s) / acceleration

Now we can substitute the value of acceleration and solve for time:

Time = (0 m/s - 8.9 m/s) / (765 N / 460 kg)

Calculating the time:

Time = -8.9 m/s / (765 N / 460 kg)

Next, we can use the formula for distance traveled during constant acceleration:

Distance = Initial velocity × time + (1/2) × acceleration × time^2

The initial velocity is 8.9 m/s, the time is the value we calculated earlier, and the acceleration is still the same. Substituting the values:

Distance = 8.9 m/s × time + (1/2) × acceleration × time^2

Now we can calculate the distance:

Distance = 8.9 m/s × (time) + (1/2) × acceleration × (time)^2

Substituting the value of time:

Distance = 8.9 m/s × ((-8.9 m/s / (765 N / 460 kg))) + (1/2) × acceleration × ((-8.9 m/s / (765 N / 460 kg)))^2

After calculating these values, you will have the distance the boat coasts (in meters).

For part (b), you already have the time it takes for the boat to come to rest after the engine is turned off. You can use the value you calculated earlier for time.