A Criss Craft Racing Runabout boat can travel at speeds up to 52 mph relative to the water. The boat is trying to reach a destination on a river that is S48°E of its point of departure. If the current is traveling at 6 knots in a southward direction, what compass reading should the boat maintain to reach its destination? How fast is the boat actually traveling relative to the shore? Also, if the destination is 50 miles from the point of departure, how long will it take the boat to reach its destination? (1 knot =1.15077945 mph).
First, we need to convert the current speed from knots to mph.
6 knots * 1.15077945 mph/knot ≈ 6.904677 mph
Now, we can analyze the problem using vector addition. The components of the Criss Craft Racing Runabout boat and the components of the current will add up to the components of the resulting motion of the boat relative to the shore.
Let the boat speed in the x-direction be Bx, and its speed in the y-direction be By. The current speed in the x-direction is Cx = 0 (since it's in the southward direction), and the current speed in the y-direction is Cy = -6.904677 mph (negative y-direction).
The destination has an angle of S48°E, which means it forms a 48-degree angle with the y-axis in the southeast quadrant. We can relate the boat's x and y components to its speed using trigonometric functions.
Bx = boat speed * cos(90 - 48)°
Bx = 52 mph * cos(42)°
Bx ≈ 39.20 mph
By = boat speed * sin(90 - 48)°
By = 52 mph * sin(42)°
By ≈ 34.55 mph
Now, we can add the boat's components and the current's components to find the components of the boat's motion.
Rx = Bx + Cx = 39.20 mph
Ry = By + Cy = 34.55 mph - 6.904677 mph ≈ 27.65 mph
We can now find the resultant speed and the angle of the boat's motion.
Resultant speed = √(Rx^2 + Ry^2) ≈ √(39.20^2 + 27.65^2) ≈ 47.67 mph
Theta = arctan(Ry / Rx) ≈ arctan(27.65 / 39.20) ≈ 34.89°
Since the speed is in the southeast quadrant, the compass reading for the boat should be S(90 - 34.89)°E ≈ S55.11°E
The boat is actually traveling at around 47.67 mph relative to the shore.
Now, we can determine the time it will take to reach its destination
time = distance / speed = 50 miles / 47.67 mph ≈ 1.049 hours or approximately 1 hour and 3 minutes.
To determine the compass reading the boat should maintain, we need to consider the effect of both the boat's velocity and the current on its heading.
Step 1: Determine the boat's velocity relative to the shore:
To find the boat's velocity relative to the shore, we need to consider the boat's velocity relative to the water and the current's velocity.
The boat's velocity relative to the shore is the vector sum of its velocity relative to the water and the current's velocity, based on their given directions.
The boat's velocity relative to the water is 52 mph, and the current's velocity is 6 knots in a southward direction. To convert the current velocity from knots to mph (as the boat's velocity is given in mph), we multiply by the conversion factor:
6 knots * 1.15077945 mph/knot = 6.9046767 mph
Now, let's calculate the boat's velocity relative to the shore:
Boat's velocity = Boat's velocity relative to the water + Current's velocity
Boat's velocity = 52 mph (relative to the water) + 6.9046767 mph (current's velocity)
Boat's velocity = 58.9046767 mph
So, the boat's velocity relative to the shore is approximately 58.9046767 mph.
Step 2: Determine the compass reading:
To determine the compass reading, we need to consider the destination's direction relative to the boat's point of departure, as well as the current's southward direction.
The destination is S48°E relative to the boat's point of departure. Since the boat is starting from the point of departure and aiming for the destination, it needs to compensate for the current's southward flow by adjusting its compass heading.
To maintain a straight path towards the destination, the boat should steer slightly to the left (east). We can calculate the amount by which it should compensate using trigonometry.
Let's consider the angle between the current's southward direction and the direction towards the destination (S48°E):
Angle = 180° - (90° + 48°)
Angle = 180° - 138°
Angle = 42°
Since the boat needs to maintain a compass heading 42° to the left (east) of the destination's direction, the compass reading should be 138°.
So, the boat should maintain a compass heading of 138° to reach its destination.
Step 3: Determine the boat's actual speed relative to the shore:
The boat's actual speed relative to the shore is the magnitude of its velocity relative to the shore that we calculated earlier, which is approximately 58.9046767 mph.
So, the boat is actually traveling at a speed of approximately 58.9046767 mph relative to the shore.
Step 4: Determine the time to reach the destination:
To calculate the time it will take the boat to reach its destination, we can divide the distance by the boat's actual speed relative to the shore.
The distance between the point of departure and the destination is given as 50 miles.
Time = Distance / Speed
Time = 50 miles / 58.9046767 mph
Time ≈ 0.8493 hours (rounded to 4 decimal places)
So, it will take the boat approximately 0.8493 hours to reach its destination, which is equivalent to about 50.96 minutes (rounded to 2 decimal places).
To determine the compass reading the boat should maintain to reach its destination, we need to consider the boat's actual speed and direction relative to the water, as well as the current's speed and direction.
1. Boat's actual speed: The boat's maximum speed is given as 52 mph relative to the water. However, we need to account for the current's speed, which is 6 knots (6 * 1.15077945 = 6.9046767 mph) in a southward direction. Since the current is southward, it opposes the boat's path. Therefore, the boat's actual speed relative to the shore will be the difference between its speed relative to the water and the current's speed. So, the boat's actual speed relative to the shore is 52 mph - 6.9046767 mph = 45.0953233 mph.
2. Compass reading: Now, let's determine the compass reading the boat should maintain. The destination is S48°E of the point of departure, which means the boat needs to follow a compass bearing that is 48° eastward from the south. However, we need to adjust this bearing based on the current's direction, which is southward. Since the current opposes the boat's path, the boat needs to maintain a compass reading slightly west of the desired bearing to counteract the current's effect. Therefore, the boat should maintain a compass reading of S48°E - westward adjustment.
3. To calculate the westward adjustment, we need to find the component of the current's speed that acts perpendicular to the desired bearing. That component can be found using trigonometry (since the desired bearing and current's direction are not orthogonal). The southward component of the current is 6 knots, and the total current speed (magnitude) is 6 knots. Let's denote the westward adjustment as X. Then, we have:
tan X = southward component / total current speed
tan X = 6 knots / 6 knots
tan X = 1
X = arctan(1)
X ≈ 45°
Hence, the boat should maintain a compass reading of S48°E - 45°, which is approximately South.
4. Time to reach the destination: The boat's actual speed relative to the shore is 45.0953233 mph. The distance to the destination is given as 50 miles. To calculate the time, we can use the formula:
time = distance / speed
Substituting the values, we have:
time = 50 miles / 45.0953233 mph
time ≈ 1.10916949 hours
Therefore, it will take approximately 1.10916949 hours (or 1 hour, 6 minutes, and 33 seconds) for the boat to reach its destination.
To summarize:
- The boat should maintain a compass reading of South.
- The boat is actually traveling at a speed of 45.0953233 mph relative to the shore.
- It will take approximately 1.10916949 hours to reach the destination.