a man can row a boat at 5km/hr. in still water and waits to cross a river to a position exactly opposite his starting point. If the river is 5km wide and is flowing at 4km/hr. eastwards find by scale drawing the direction in which he must set off in order to accomplish his objective. How long will it take to cross the river
To determine the direction in which the man must set off, we can draw a vector diagram to represent the velocity of the boat and the velocity of the river.
Let's assume the direction that the man wants to travel is represented by the vector A, and the velocity of the river is represented by the vector B. We can draw vector A going directly from the starting point to the point opposite, and vector B going directly eastward.
Using the given information, we know that the velocity of the boat relative to still water is 5 km/hr, but the river is flowing eastward at a velocity of 4 km/hr. To determine the actual velocity of the boat in the river, we need to find the resultant of the boat's velocity and the river's velocity.
We can use vector addition to find the resultant velocity. Since the boat wants to go directly across the river, the angle between the boat's velocity and the river's velocity will be 90 degrees.
Using the Pythagorean theorem, we can find the magnitude of the resultant velocity:
Resultant_velocity = √(5^2 + 4^2)
= √(25 + 16)
= √41
Now, let's draw vector A with a magnitude of √41, going from the starting point to the opposite point, making a right angle with vector B. This will represent the direction the man must set off in order to cross the river.
To find the time it will take to cross the river, we need to determine the distance the boat needs to travel. The width of the river is given as 5 km.
Since the man wants to reach a point exactly opposite his starting point, the distance he needs to travel across the river is also 5 km.
Now, we can use the formula for time:
Time = Distance / Speed
Time = 5 km / √41 km/hr
= 5 / √41 hours
To simplify the answer, we can multiply the numerator and denominator by √41:
Time = (5 * √41) / (√41 * √41)
= 5√41 / 41
Therefore, it will take the man approximately 5√41 / 41 hours to cross the river.
To determine the direction in which the man must set off and calculate the time it will take to cross the river, we can use vector addition.
1. Start by drawing a scale diagram with the river flowing eastwards. Mark the man's starting position on the western bank of the river.
2. Draw a line perpendicular to the river, representing the man's instantaneous velocity relative to the water. This line should have a length of 5 km, as the man can row at a speed of 5 km/hr in still water.
3. Next, draw a vector representing the velocity of the river, flowing eastwards. This vector should have a length of 4 km, since the river is flowing at a speed of 4 km/hr.
4. Now, using vector addition, draw the resultant vector by adding the man's velocity vector and the river's velocity vector. The resultant vector represents the direction and magnitude of the man's actual velocity.
5. The direction of the resultant vector represents the direction in which the man must set off to cross the river directly opposite his starting point.
6. Measure the length of the resultant vector on the scale drawing. This will give you an idea of the magnitude of the man's actual velocity.
7. To calculate the time it takes to cross the river, divide the width of the river (5 km) by the magnitude of the man's actual velocity (determined from the scale drawing). This will give you the time in hours.
Note: This method assumes that the man rows perpendicular to the river's flow to reach the opposite shore.
draw the velocity vectors. They form a 3-4-5 right triangle, with the straight-across side of length 3.
So, scale it up by a factor of 5/3 to get an equivalent distance triangle. He will have to row 25/3 km along his path, while drifting downstream 20/3 km, landing exactly 5 km from where he started.