To determine the magnitude of the acceleration of the boat, we need to use the initial and final velocities, the time interval, and the concept of vector subtraction. Here are the steps:
1. **Convert Initial and Final Velocities to Vector Components:**
\[
v_{0x} = v_0 \cos(\theta_0)
\]
\[
v_{0y} = v_0 \sin(\theta_0)
\]
\[
v_{fx} = v_f \cos(\theta_f)
\]
\[
v_{fy} = v_f \sin(\theta_f)
\]
Given:
\[
v_0 = 13.6 \, \text{m/s}, \quad \theta_0 = 74.0^\circ
\]
\[
v_f = 17.1 \, \text{m/s}, \quad \theta_f = 79.8^\circ
\]
Using trigonometric identities:
\[
v_{0x} = 13.6 \cos(74.0^\circ)
\]
\[
v_{0y} = 13.6 \sin(74.0^\circ)
\]
\[
v_{fx} = 17.1 \cos(79.8^\circ)
\]
\[
v_{fy} = 17.1 \sin(79.8^\circ)
\]
2. **Calculate Initial and Final Components:**
\[
v_{0x} = 13.6 \cos(74.0^\circ) \approx 13.6 \cdot 0.2756 \approx 3.75 \, \text{m/s}
\]
\[
v_{0y} = 13.6 \sin(74.0^\circ) \approx 13.6 \cdot 0.9613 \approx 13.07 \, \text{m/s}
\]
\[
v_{fx} = 17.1 \cos(79.8^\circ) \approx 17.1 \cdot 0.1720 \approx 2.94 \, \text{m/s}
\]
\[
v_{fy} = 17.1 \sin(79.8^\circ) \approx 17.1 \cdot 0.9851 \approx 16.85 \, \text{m/s}
\]
3. **Find the Changes in Velocity Components (Δv_x and Δv_y):**
\[
\Delta v_x = v_{fx} - v_{0x} = 2.94 \, \text{m/s} - 3.75 \, \text{m/s} = -0.81 \, \text{m/s}
\]
\[
\Delta v_y = v_{fy} - v_{0y} = 16.85 \, \text{m/s} - 13.07 \, \text{m/s} = 3.78 \, \text{m/s}
\]
4. **Find the Magnitude of the Change in Velocity (Δv):**
\[
\Delta v = \sqrt{(\Delta v_x)^2 + (\Delta v_y)^2} = \sqrt{(-0.81)^2 + (3.78)^2}
\]
\[
\Delta v = \sqrt{0.6561 + 14.2884} = \sqrt{14.9445} \approx 3.87 \, \text{m/s}
\]
5. **Calculate the Magnitude of the Acceleration (a):**
\[
a = \frac{\Delta v}{\Delta t}
\]
Given the time interval \( \Delta t = 2.62 \, \text{s} \):
\[
a = \frac{3.87 \, \text{m/s}}{2.62 \, \text{s}} \approx 1.478 \, \text{m/s}^2
\]
Therefore, the magnitude of the acceleration of the boat is approximately \(1.478 \, \text{m/s}^2\).
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
1 answer