Suppose you would like to cross a 209-foot wide river in a boat. Assume that the boat can travel 32 mph relative to the water and that the current is flowing west at the rate of 6 mph. If the bearing chosen is chosen so that the boat will land at a point exactly across from its starting loin, how long will it take for the boat to make the crossing? Give your answer to the nearest second.
if the boat travels s seconds,
1 km/hr = 1.4667 ft/s
209^2 + (6*1.4667s)^2 = (32*1.4667s)^2
s = 4.5 sec
1 mile/hr=1.46667 ft/sec. The KM here is a typo. The rest o the problem is good and correct. Giving direction on how to draw a diagram to visualize would have been be great.
Thanks and keep up the good work.
To solve this problem, we can use the concept of vector addition.
Let's break down the velocity vectors into their components. In this case, we have the following:
Boat velocity in the x-direction (east): 32 mph
Current velocity in the y-direction (north): 6 mph
Now, we can use the Pythagorean theorem to find the resultant velocity of the boat:
Resultant velocity = √((boat velocity)^2 + (current velocity)^2)
= √((32 mph)^2 + (6 mph)^2)
≈ √(1024 mph^2 + 36 mph^2)
≈ √1060 mph^2
≈ 32.62 mph
The boat will be traveling at a velocity of approximately 32.62 mph in a direction that will allow it to land exactly across from its starting point.
Next, we can calculate the time it takes for the boat to cross the river using the formula:
Time = Distance / Velocity
The distance to be crossed is 209 feet, and the velocity is 32.62 mph. However, we need to convert the units to be consistent:
Distance = 209 feet
Velocity = 32.62 mph * (5280 feet/1 mile) * (1 hour/60 minutes)
≈ 1514.16 feet/minute
Now, we can calculate the time:
Time = 209 feet / 1514.16 feet/minute
≈ 0.1380 minutes
To convert minutes to seconds, we can multiply by 60:
Time ≈ 0.1380 minutes * 60 seconds/minute
≈ 8.28 seconds
Therefore, it will take approximately 8.28 seconds for the boat to make the crossing.
To calculate the time it will take for the boat to cross the river, we need to use the concept of relative velocity.
Here's how you can calculate it step-by-step:
1. Start by breaking down the boat's velocity into its horizontal and vertical components.
Velocity of the boat in still water (Vbs) = 32 mph
Velocity of the river current (Vr) = 6 mph
Since the current is flowing westward, the horizontal component of the boat's velocity is unaffected by the current, and the vertical component is given by the current velocity.
Horizontal component (Vbx) = 32 mph
Vertical component (Vby) = 6 mph
2. Calculate the resultant velocity of the boat relative to the ground by using vector addition.
Relative velocity of the boat (Vb) = √((Vbx)^2 + (Vby)^2)
Vb = √((32 mph)^2 + (6 mph)^2)
Vb ≈ 32.55 mph
3. Determine the time it will take for the boat to cross the river by using the formula:
Time = Distance / Relative Velocity
Distance = 209 feet
Relative Velocity = 32.55 mph
Convert the distance from feet to miles: 209 feet ÷ 5280 feet/mile ≈ 0.0396 miles.
Time = 0.0396 miles / 32.55 mph
4. Solve for time:
Time ≈ 0.0012 hours
Convert hours to seconds: 0.0012 hours × 3600 seconds/hour ≈ 4.32 seconds
Therefore, it will take the boat approximately 4 seconds to make the crossing.