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Chapter 4: Problem 11

A boat moves through the water with two forces acting on it. One is a \(2000-\mathrm{N}\) forward push by the water on the propeller, and the other is a \(1800-\mathrm{N}\) resistive force due to the water around the bow. (a) What is the acceleration of the \(1000-\mathrm{kg}\) boat? (b) If it starts from rest, how far will the boat move in \(10.0 \mathrm{~s}\) ? (c) What will its velocity be at the end of that time?

Short Answer

Expert verified
The acceleration of the boat is \(0.2 m/s^{2}\), it will move a distance of 10m in 10 seconds, and its velocity at the end of that time will be \(2 m/s\).

Step by step solution

01

- Calculate the Net Force on the Boat

From Newton's second law, the net force on an object is equal to the mass of the object multiplied by its acceleration. But first, we need to calculate the net force. The net force acting on the boat is the difference between the forward push and the resistive force. This can be obtained by subtracting the resistive force from the forward push: \(F_{net} = F_{forward} - F_{resistive} = 2000N - 1800N = 200N \). So, the net force acting on the boat is 200N in the direction of the forward push.
02

- Find the Acceleration

Now that the net force has been found, the acceleration (a) can be calculated by rearranging Newton's second law: \(a = \frac{F_{net}}{m} = \frac{200N}{1000kg} = 0.2 m/s^{2}\). The acceleration of the boat is 0.2 m/s虏 towards the forward direction.
03

- Calculate Displacement using Equations of Motion

Now that the acceleration of the boat is known, we can find the displacement (distance travelled) using the equations of motion. Since the boat starts from rest, the initial velocity \(u = 0 m/s\) . We apply the second equation of motion: \(s = ut + \frac{1}{2}at^{2} = 0\times10s + 0.5 \times 0.2 m/s^{2} \times (10s)^{2} = 10m\). The boat will move 10m in 10s.
04

- Find the Final Velocity

The final velocity can be calculated using the first equation of motion: \(v = u + at = 0 m/s + (0.2 m/s^{2})(10s) = 2 m/s\) . Therefore, the velocity of the boat at the end of 10 s will be 2 m/s in the forward direction.

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