A swimmer is capable of swimming at 1.4m/s in still water. Calculate the following.
i. How far downstream will he land if he swims directly across 180m wide river?
ii. How long will it take him to the other side.
To answer these questions, we need to consider the swimmer's velocity relative to the ground and the river's current.
Let's assume the river's current has a speed of 0.6 m/s downstream.
i. To calculate how far downstream the swimmer will land, we need to determine the component of the swimmer's velocity perpendicular to the river's flow. This component will allow us to calculate the distance.
The swimmer's velocity relative to the ground can be found using the Pythagorean theorem:
v_ground = sqrt(v_swimmer^2 - v_current^2)
= sqrt((1.4 m/s)^2 - (0.6 m/s)^2)
= sqrt(1.96 - 0.36)
= sqrt(1.6)
= 1.26 m/s (approximately)
Since the swimmer is swimming directly across the river, the downstream distance he will land is equal to the time taken multiplied by his velocity relative to the ground:
distance = time * v_ground
To find the time taken, we divide the width of the river (180 m) by the swimmer's velocity relative to the ground:
time = distance / v_ground
= 180 m / 1.26 m/s
= 142.86 s (approximately)
Therefore, the swimmer will land approximately 142.86 meters downstream.
ii. To calculate the time it will take for the swimmer to reach the other side, we divide the width of the river (180 m) by the swimmer's velocity in still water:
time = distance / v_swimmer
= 180 m / 1.4 m/s
= 128.57 s (approximately)
Therefore, it will take the swimmer approximately 128.57 seconds to reach the other side of the river.
To calculate the distance downstream where the swimmer will land and the time it will take him to reach the other side of the river, we first need to break down the problem into its components.
i. Distance downstream:
The swimmer's speed in still water is 1.4 m/s. However, since there is a river flowing and that will affect the swimmer's movement, we need to determine the swimmer's effective speed when swimming across the river.
Let's assume that the current of the river flows at a speed of v m/s. In this case, the swimmer's effective speed will be the vector sum of his speed in still water and the speed of the river's current.
To calculate the effective speed, we use the Pythagorean theorem:
effective speed = √((swimmer's speed)^2 + (river's speed)^2)
In this case, the swimmer's speed is 1.4 m/s, and since the river's current is perpendicular to the swimmer's direction, the river's speed will be equal to the swimmer's speed downstream (v = 1.4 m/s).
Therefore, the effective speed will be:
effective speed = √((1.4 m/s)^2 + (1.4 m/s)^2)
effective speed = √3.92 m^2/s^2
effective speed ≈ 1.98 m/s (approximated to two decimal places)
Now, using the effective speed, we can calculate the distance downstream (d) where the swimmer will land:
distance downstream = effective speed × time taken to cross the river
ii. Time to cross the river:
To calculate the time, we need to know the width of the river and the swimmer's effective speed.
Given that the river is 180 meters wide and the swimmer's effective speed is 1.98 m/s, we can use the formula:
time = distance / speed
In this case, the distance is 180 meters and the speed is 1.98 m/s.
Now that we have the necessary information and formulas, we can calculate the required values.
i. Distance downstream:
distance downstream = effective speed × time taken to cross the river
(Using the calculated effective speed)
distance downstream = 1.98 m/s × time taken to cross the river
ii. Time to cross the river:
time = distance / speed
(Using the given distance of 180 meters and the calculated effective speed)
time = 180 meters / 1.98 m/s
Let's calculate the values.
i. what is the speed of the current in the river?
ii. (180 m) / (1.4 m/s)