A boater wants to head straight across a river that has a current of 3 km/hr. If the top speed of the boat in still water is 4 km/hr. 1.find the direction the boat must travel into the current to go straight across the river?
2. How fast is the boat going straight across the river?
Another boat with a bigger motor wants to cross the same river at the same point but wants to travel at a constant 4 km/hr rate.
3. What angle does the boat head into the current and how fast is the boat going with respect to the water around it?
1. Vr = Vc + Vb = 4.
-3i + Vb = = 4, Vb = 4 + 3i = 5km/h[36.9o].
Direction = 36.9o N. of E.
2. Vr = 4 km/h. = Resultant velocity.
3. 5km/h[36.9o].
To answer these questions, we need to use the concept of vector addition. Here's how you can find the solutions step by step:
1. Find the direction the boat must travel into the current:
To go straight across the river, the boat must offset the effect of the current. Since the current is flowing to the right, the boat needs to point slightly to the left to counteract it. To calculate this angle, we can use the tangent function:
angle = arctan(current speed / boat speed)
= arctan(3/4)
Using a calculator, the approximate angle is 36.87 degrees. Therefore, the boat must travel at an angle of approximately 36.87 degrees to the left of the straight path across the river.
2. Find the speed of the boat when traveling straight across the river:
To find the speed of the boat, we can use the Pythagorean theorem. The speed of the boat when traveling straight across the river is the hypotenuse of a right-angled triangle, where the two other sides represent the boat's speed in still water and the speed of the current:
speed^2 = (boat speed)^2 + (current speed)^2
speed^2 = 4^2 + 3^2
speed^2 = 16 + 9
speed^2 = 25
speed = √25
speed = 5 km/hr
Therefore, the boat's speed when traveling straight across the river is 5 km/hr.
3. Find the angle and speed of the larger boat:
Since the boat wants to travel at a constant 4 km/hr rate, it needs to overcome the current completely. This means that the boat must point directly across the river, perpendicular to the current. The angle at which the boat should head into the current is 90 degrees.
The speed of the boat with respect to the water around it will be the same as its speed in still water, which is 4 km/hr. Since it is moving perpendicular to the current, the current does not affect the boat's speed in this direction.
To summarize:
1. The boat must travel at an angle of approximately 36.87 degrees to the left of the straight path across the river.
2. The boat's speed when traveling straight across the river is 5 km/hr.
3. The larger boat should head into the current at a 90-degree angle and will maintain a speed of 4 km/hr with respect to the water around it.