the water in a 0.5 km wide river is flowing at 5 m/sec from west to east. a man row a boat with a velocity of 13 m/sec in a direction of north west relative to water. find the time taken to cross the river so that man reaches exactly at same point to opposite end of the river.
Ans is : 41.67 sec . How??
Good
To solve this problem, we can break down the motion of the boat into two components: the velocity due to the river's flow and the velocity of the boat relative to the water.
Let's assume the width of the river is "d" and the time taken to cross the river is "t".
1. The velocity due to the river's flow can be calculated using the formula v = d/t, where v is the velocity and d is the distance.
In this case, the velocity due to the river's flow is 5 m/sec (from west to east) and the width of the river is 0.5 km = 500 m.
Therefore, the time taken to cross the river is t = d/v = 500/5 = 100 seconds.
2. Now, we need to calculate the velocity of the boat relative to the water. The boat is rowing at a velocity of 13 m/sec in a north-west direction. To find the velocity of the boat relative to the water, we need to resolve this velocity into its northward and westward components.
The northward component will be the vertical component of the velocity, given by v_north = v * cos(45°) = 13 * cos(45°) = 9.19 m/sec.
The westward component will be the horizontal component of the velocity, given by v_west = v * sin(45°) = 13 * sin(45°) = 9.19 m/sec.
3. Now, we can use the relative velocity of the boat and the time taken to cross the river to determine the distance covered by the boat.
The distance covered in the northward direction will be given by d_north = v_north * t = 9.19 * t.
The distance covered in the westward direction due to the river's flow will be given by d_west = 5 * t.
4. Since the boat wants to reach exactly the opposite end of the river, the distance covered in the northward direction should be equal to the distance covered in the westward direction. Therefore, we can equate the two distances.
Equating the distances, we get 9.19 * t = 5 * t.
Simplifying, we find t = 0.54 seconds.
Therefore, the time taken to cross the river so that the man reaches exactly at the same point on the opposite end of the river is approximately 0.54 seconds.
(rounded to two decimal places, the answer is not 41.67 seconds as stated in the question)