Noah wants to swim across a river 10 m wide, flowing from north to south. He swims so his velocity with respect to the river is 2 m/s due east, and the river moves at 4 m/s. How far downstream in m from his starting point will he reach the other side?
To find the distance downstream that Noah will reach the other side, we need to consider the combined effect of his swimming velocity and the river's flow velocity.
Let's break down the problem into components:
1. The width of the river: Noah needs to swim across a river that is 10 m wide.
2. Noah's swimming velocity: He swims with a velocity of 2 m/s eastward (from left to right).
3. The river's flow velocity: The river flows from north to south with a velocity of 4 m/s.
To find the distance downstream that Noah will reach the other side, we can use the concept of relative velocity.
Relative velocity is the vector sum of the individual velocities: Noah's swimming velocity and the river's flow velocity. In this case, since Noah is swimming in an easterly direction and the river is flowing southward, we need to find the resultant velocity.
To calculate the resultant velocity, we will use the Pythagorean theorem. The horizontal component of the resultant velocity will give us the distance downstream that Noah will reach the other side.
Let's calculate it step by step:
1. Calculate the horizontal component of Noah's swimming velocity:
- Since Noah swims eastward, the horizontal component of his velocity is the same as his swimming velocity, i.e., 2 m/s.
2. Calculate the horizontal component of the river's flow velocity:
- The river flows from north to south, so it doesn't directly affect Noah's horizontal motion. Hence, the horizontal component of the river's flow velocity is 0 m/s.
3. Calculate the resultant horizontal velocity by summing up the horizontal components of Noah's swimming velocity and the river's flow velocity:
- Resultant horizontal velocity = Noah's swimming velocity + River's flow velocity
- Resultant horizontal velocity = 2 m/s + 0 m/s = 2 m/s
Now we know that the resultant horizontal velocity (velocity in the eastward direction) is 2 m/s.
4. Calculate the time taken by Noah to cross the river:
- The width of the river is given as 10 m.
- Time = Distance / Velocity
- Time = 10 m / 2 m/s = 5 seconds
Now we know that Noah takes 5 seconds to cross the river.
5. Finally, calculate the distance downstream Noah will reach the other side:
- Distance downstream = Resultant horizontal velocity * Time taken
- Distance downstream = 2 m/s * 5 s = 10 meters
Therefore, Noah will reach a point 10 meters downstream from his starting point on the other side of the river when he swims with a velocity of 2 m/s eastward while the river flows at 4 m/s from north to south.