The width of river is 1 km. The velocity of boat is 5 km/hr. The boat covered the width of river with shortest will possible path in 15 min. Then the velocity of river stream is:

Question

A ball is thrown vertically upwards with an initial velocity of $ 20 \, \text{m/s} $. How high will the ball rise? (Take $ g = 10 \, \text{m/s}^2 $)

Answer 1

To determine the velocity of the river stream, let's analyze the given situation and use the formulae of relative motion in water.

Given:

Width of the river (distance to be covered): 1 \ \text{km}
Velocity of the boat with respect to still water: 5 \ \text{km/hr}
Time taken to cross the river: 15 \ \text{min} = \frac{15}{60} \ \text{hr} = 0.25 \ \text{hr}

Since the boat takes the shortest possible path, the boat needs to move perpendicular to the flow of the river. Therefore, the time taken to traverse across the river is given by:

t = \frac{\text{width of river}}{\text{velocity of boat across river}}

From the given data, we know:

0.25 = \frac{1}{V_{\perp}}

Solving for V_{\perp}:

V_{\perp} = \frac{1}{0.25} = 4 \ \text{km/hr}

Since the velocity of the boat in still water is 5 \ \text{km/hr}, we use the Pythagorean theorem to find the velocity of the river stream V_r. The components are related by:

(V_{\perp})^2 + (V_r)^2 = (5)^2

We have V_{\perp} = 4 \ \text{km/hr}, so:

4^2 + (V_r)^2 = 5^2

16 + (V_r)^2 = 25

(V_r)^2 = 9

V_r = \sqrt{9} = 3 \ \text{km/hr}

Conclusion: The velocity of the river stream is 3 \ \text{km/hr}, which confirms the correct answer: 3 \ \text{km/hr}.

Answer 2