6. A boat steers across a river at 15m/s, the current flows at 5.0m/s What is the resultant velocity? 7. How long does it take to cross the river If the river in #6 is 40.0 m-wide? 8. How far downstream n does the boat end up in #7? gift Theta =13.4^circ 9. A boat wants to get directly across a stream:the engine is capable of attaining a speed of 13.0 m/s. If the current is 5.0m/s at what angle should the boat travel a nd what would its speed be as Ask viewed from shore? 10. An airplane flies at 200m/s at 45^circ E of N, a wind blows at 75m/s at 30^circ W of N. What is the resultant velocity of the aircraft as seen from the ground? 11. Two boys pull on ropes tied to a rock by the following forces 150 Nat 30^circ E of S, 160 N at 20^circ W of S. Sketch vectors to represent this and calculate the net force acting on the stump. 12. If the stump in the above question does not move what is the minimum force of friction (magnitude and direction)in this situation?

6. **Resultant Velocity:** We use the Pythagorean theorem since the boat's velocity and the river current are perpendicular.<br /><br />Resultant velocity = √((15 m/s)² + (5 m/s)²) = √(225 + 25) = √250 ≈ **15.8 m/s**<br /><br />7. **Time to Cross:** The time to cross depends only on the boat's velocity component perpendicular to the current (15 m/s) and the river width.<br /><br />Time = Distance / Velocity = 40.0 m / 15 m/s ≈ **2.67 s**<br /><br />8. **Distance Downstream:** The distance downstream depends on the current's velocity and the time it takes to cross.<br /><br />Distance downstream = Current velocity * Time = 5.0 m/s * 2.67 s ≈ **13.4 m**<br /><br />9. **Angle and Speed to Cross Directly:**<br /><br />* **Angle:** Let θ be the angle upstream the boat should head. The boat's velocity component against the current must equal the current's velocity.<br /> 13.0 m/s * sin(θ) = 5.0 m/s<br /> sin(θ) = 5.0/13.0<br /> θ = arcsin(5.0/13.0) ≈ **22.6°** upstream (or North of West if the stream flows West to East).<br /><br />* **Speed from Shore:** The boat's velocity component across the stream is 13.0 m/s * cos(θ). This is the speed observed from the shore.<br /> Speed from shore = 13.0 m/s * cos(22.6°) ≈ **12.0 m/s**<br /><br />10. **Resultant Aircraft Velocity:** We need to break down the velocities into their North and East components and then add them.<br /><br />* **Airplane:**<br /> * North component: 200 m/s * cos(45°) ≈ 141.4 m/s<br /> * East component: 200 m/s * sin(45°) ≈ 141.4 m/s<br /><br />* **Wind:**<br /> * North component: 75 m/s * cos(30°) ≈ 65.0 m/s<br /> * West component: 75 m/s * sin(30°) ≈ -37.5 m/s (negative since it's West)<br /><br />* **Resultant:**<br /> * North component: 141.4 m/s + 65.0 m/s = 206.4 m/s<br /> * East component: 141.4 m/s - 37.5 m/s = 103.9 m/s<br /><br />* **Magnitude:** √((206.4 m/s)² + (103.9 m/s)²) ≈ **230.7 m/s**<br />* **Direction:** arctan(103.9/206.4) ≈ 26.7° East of North<br /><br /><br />11. **Net Force on Rock:**<br /><br />* **Sketch:** Draw two vectors originating from the same point. One vector at 30° East of South (150 N) and the other at 20° West of South (160 N).<br /><br />* **Calculation:** Resolve each force into South and East/West components:<br /> * 150 N force:<br /> * South: 150 N * cos(30°) ≈ 129.9 N<br /> * East: 150 N * sin(30°) = 75 N<br /> * 160 N force:<br /> * South: 160 N * cos(20°) ≈ 150.4 N<br /> * West: 160 N * sin(20°) ≈ -54.7 N (negative for West)<br /><br />* **Net Force:**<br /> * South: 129.9 N + 150.4 N = 280.3 N<br /> * East/West: 75 N - 54.7 N = 20.3 N East<br /><br />* **Magnitude:** √((280.3 N)² + (20.3 N)²) ≈ **281.1 N**<br />* **Direction:** arctan(20.3/280.3) ≈ 4.1° East of South<br /><br />12. **Friction Force:** If the stump doesn't move, the friction force must exactly oppose the net force calculated in #11.<br /><br />* **Magnitude:** **281.1 N**<br />* **Direction:** **4.1° West of North** (opposite to the net pulling force)<br />