To find the magnitude of the acceleration of the boat, we need to use the change in velocity and the time over which this change occurs. Here's the step-by-step process:
1. **Initial and Final Velocities**: Given in polar form
- Initial velocity \( \mathbf{v_i} = 13.6 \, \text{m/s} \) at \( 74.0^\circ \)
- Final velocity \( \mathbf{v_f} = 17.1 \, \text{m/s} \) at \( 79.8^\circ \)
2. **Convert Polar to Rectangular Coordinates**:
\[
\mathbf{v_i} = 13.6 (\cos 74.0^\circ \, \hat{i} + \sin 74.0^\circ \, \hat{j})
\]
\[
\mathbf{v_f} = 17.1 (\cos 79.8^\circ \, \hat{i} + \sin 79.8^\circ \, \hat{j})
\]
Calculating the components:
\[
\mathbf{v_i} = 13.6 (\cos 74.0^\circ \, \hat{i} + \sin 74.0^\circ \, \hat{j}) = 13.6 (0.2756 \, \hat{i} + 0.9613 \, \hat{j}) = 3.748 \, \hat{i} + 13.067 \, \hat{j}
\]
\[
\mathbf{v_f} = 17.1 (\cos 79.8^\circ \, \hat{i} + \sin 79.8^\circ \, \hat{j}) = 17.1 (0.1794 \, \hat{i} + 0.9838 \, \hat{j}) = 3.067 \, \hat{i} + 16.819 \, \hat{j}
\]
3. **Calculate the Change in Velocity**:
\[
\Delta \mathbf{v} = \mathbf{v_f} - \mathbf{v_i}
\]
\[
\Delta \mathbf{v} = (3.067 \, \hat{i} + 16.819 \, \hat{j}) - (3.748 \, \hat{i} + 13.067 \, \hat{j}) = (-0.681 \, \hat{i}) + (3.752 \, \hat{j})
\]
4. **Magnitude of the Change in Velocity**:
\[
|\Delta \mathbf{v}| = \sqrt{(-0.681)^2 + (3.752)^2} = \sqrt{0.463 + 14.081} = \sqrt{14.544} \approx 3.812 \, \text{m/s}
\]
5. **Calculate the Magnitude of the Acceleration**:
\[
\mathbf{a} = \frac{\Delta \mathbf{v}}{\Delta t}
\]
\[
a = \frac{|\Delta \mathbf{v}|}{\Delta t} = \frac{3.812}{2.62} \approx 1.455 \, \text{m/s}^2
\]
Therefore, the magnitude of the acceleration of the boat is approximately \(1.455 \, \text{m/s}^2\).
a sailboat is moving at 13.6 m/s at an angle of 74.0 degrees when a wind blows it for 2.62s. after, the boat moves with a velocity of 17.1 m/s in an 79.8 degrees angle. What is the magnitude of the acceleration of the boat?
a=[?] m/s^2
remember to add 180 degrees when necessary
1 answer
1 answer