Answer:
the ship traveled approximately 1.82 kilometers during its acceleration from 14 knots to 17 knots.
Step-by-step explanation:
To determine the distance the ship traveled, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy. The kinetic energy (KE) of an object with mass (m) and velocity (v) can be calculated using the following formula:
\[KE = \frac{1}{2}mv^2\]
The change in kinetic energy (ΔKE) is equal to the work done (W) on the object:
\[ΔKE = W\]
Given that the ship's mass (m) is 20,000 kg and its initial velocity (u) is 14 knots, and final velocity (v) is 17 knots, we need to convert the velocities to meters per second (m/s) because the mass is given in kilograms.
1 knot is approximately equal to 0.514444 m/s. So, we have:
Initial velocity (u) = 14 knots = 14 * 0.514444 m/s = 7.21 m/s
Final velocity (v) = 17 knots = 17 * 0.514444 m/s = 8.74 m/s
Now, we can calculate the initial kinetic energy (KE1) and final kinetic energy (KE2) of the ship:
\[KE1 = \frac{1}{2} \times 20,000 \, \text{kg} \times (7.21 \, \text{m/s})^2\]
\[KE2 = \frac{1}{2} \times 20,000 \, \text{kg} \times (8.74 \, \text{m/s})^2\]
The change in kinetic energy (ΔKE) is given by:
\[ΔKE = KE2 - KE1\]
Now, since ΔKE is equal to the work done on the ship (W), we have:
\[W = ΔKE = KE2 - KE1\]
Substitute the values and calculate ΔKE:
\[W = \left[\frac{1}{2} \times 20,000 \, \text{kg} \times (8.74 \, \text{m/s})^2\right] - \left[\frac{1/2}{2} \times 20,000 \, \text{kg} \times (7.21 \, \text{m/s})^2\]
Now, calculate ΔKE:
\[ΔKE = W = \left[\frac{1}{2} \times 20,000 \, \text{kg} \times (8.74 \, \text{m/s})^2\right] - \left[\frac{1}{2} \times 20,000 \, \text{kg} \times (7.21 \, \text{m/s})^2\]
\[ΔKE ≈ 899,346 \, \text{Joules}\]
The work done on the ship is approximately 899,346 Joules.
Now, we can use the work-energy principle to find the distance (d) the ship traveled. The work done is equal to the change in kinetic energy:
\[W = ΔKE = \frac{1}{2}mv^2 - \frac{1}{2}mu^2\]
We already have the values for W, m, u, and v. Rearrange the equation to solve for d:
\[d = \frac{W}{\frac{1}{2}mv^2 - \frac{1}{2}mu^2}\]
Now, plug in the values:
\[d = \frac{899,346 \, \text{J}}{\frac{1}{2} \times 20,000 \, \text{kg} \times (8.74 \, \text{m/s})^2 - \frac{1}{2} \times 20,000 \, \text{kg} \times (7.21 \, \text{m/s})^2}\]
Calculate the value of d:
\[d ≈ \frac{899,346 \, \text{J}}{0.5 \times 20,000 \, \text{kg} \times (76.5156 \, \text{m^2/s^2}) - 0.5 \times 20,000 \, \text{kg} \times (51.8841 \, \text{m^2/s^2})}\]
\[d ≈ \frac{899,346 \, \text{J}}{1530.312 - 1037.682}\]
\[d ≈ \frac{899,346 \, \text{J}}{492.63}\]
\[d ≈ 1821.35 \, \text{meters} \, \text{or} \, 1.82 \, \text{kilometers}\]