A man at point A directs his rowboat due north toward point B, straight across a river of
width 100 m. The river current is due east. The man starts across, rowing steadily at 2.50
m/s and reaches the other side of the river at point C, 150 m downstream from his starting
point. While the man is crossing the river, what is his velocity relative to the shore?
1.1
To find the man's velocity relative to the shore while crossing the river, we can use vector addition.
Step 1: Determine the velocity of the river current
The man is rowing steadily at 2.50 m/s due north. Since the river current is due east, it does not affect the man's northward speed. Therefore, the velocity of the river current is only in the eastward direction, and its magnitude is unknown.
Step 2: Break down the man's velocity into components
The man's velocity can be broken down into two components: one in the north direction and one in the east direction. The northward component is 2.50 m/s, and the eastward component is unknown.
Step 3: Determine the eastward component of the man's velocity using the distance and time measurements given
The man reaches point C, which is 150 m downstream from his starting point. This means that his total displacement in the eastward direction is 150 m. The time it takes to cross the river is not given, so we cannot directly calculate the man's eastward velocity. However, since the man is rowing across the river in a straight line, we can assume he crosses the river in the shortest time possible. This means that his eastward distance traveled (150 m) is equal to the product of his eastward velocity and the time taken to cross the river.
Step 4: Use vector addition to find the man's velocity relative to the shore
To find the man's velocity relative to the shore, we need to add his northward velocity component (2.50 m/s) to his eastward velocity component. The northward component remains unchanged as there is no force in the north direction. The eastward component is calculated in step 3.
So, the man's velocity relative to the shore is the vector sum of his northward velocity component (2.50 m/s) and his eastward velocity component (150 m divided by the time taken to cross the river).
Note: The exact numerical value of the man's velocity relative to the shore cannot be determined without knowing the time taken to cross the river or the exact value of the eastward component.
To solve this problem, we can break it down into two components: the velocity of the boat in still water and the velocity of the river current.
Let's assume the velocity of the boat in still water is Vb and the velocity of the river current is Vc.
Given:
- The man rows across the river from point A to point C, which is 150 m downstream from point A.
- The width of the river is 100 m.
- The man rows at a constant speed of 2.50 m/s.
Now we can use the concept of vectors to solve the problem.
Step 1: Determine the time it takes for the man to cross the river.
We can use the formula:
Time = Distance / Speed
Time = 100 m / 2.50 m/s
Time = 40 seconds
Step 2: Find the horizontal distance the boat is being carried downstream by the current during the crossing.
The horizontal distance, which is the distance the boat is carried downstream by the current, can be found using the formula:
Distance = Velocity * Time
Since the velocity of the river current is due east, the horizontal distance is equal to the velocity of the river current multiplied by the time it takes to cross the river.
Distance = Vc * Time
Distance = Vc * 40 s
Distance = 40Vc meters
Step 3: Find the distance the man has traveled across the river.
Since the width of the river is 100 m and the man reached point C, which is 150 m downstream, the distance the man has traveled across the river is:
Distance = 150 m - 100 m
Distance = 50 meters
Step 4: Calculate the velocity of the boat in still water.
The velocity of the boat in still water can be calculated using the formula:
Velocity in still water = Distance / Time
Velocity in still water = 50 m / 40 s
Velocity in still water = 1.25 m/s
Step 5: Determine the velocity of the boat relative to the shore.
The velocity of the boat relative to the shore can be calculated by finding the resultant vector of the boat's velocity in still water and the velocity of the river current.
Using vector addition, we can find the resultant velocity using the Pythagorean theorem:
Resultant Velocity^2 = (Velocity in still water)^2 + (Velocity of the river current)^2
Resultant Velocity = sqrt((Velocity in still water)^2 + (Velocity of the river current)^2)
Resultant Velocity = sqrt((1.25 m/s)^2 + (Vc)^2)
Since we don't have the value of Vc, we cannot calculate the exact velocity of the boat relative to the shore. However, we do know that it is the magnitude (or the length) of the resultant vector.
In conclusion, we need to know the value of the river current (Vc) to determine the velocity of the boat relative to the shore.