On a rough day there is a uniform current in the river,the speed of da river is v .which will help the onward journey and impede the return.if the boat takes t sec to go and come back .wat is ratio between to/t

To tackle the problem of a boat traveling in a river with a current, we need to break down the journey into two parts: the onward journey downstream and the return journey upstream. The current of the river plays a crucial role in determining the effective speed of the boat in each direction.

Understanding the Journey

Let's define some variables to make our calculations clearer:

• v: Speed of the river current
• b: Speed of the boat in still water
• t: Total time taken for the round trip
• t_to: Time taken to go downstream
• t_back: Time taken to return upstream

Effective Speeds

When the boat is moving downstream, the effective speed is the sum of the boat's speed and the river's current:

Effective speed downstream = b + v

Conversely, when the boat is moving upstream, the effective speed is the boat's speed minus the river's current:

Effective speed upstream = b - v

Time Calculations

Let’s say the distance traveled downstream (and upstream) is d. The time taken to go downstream can be expressed as:

t_to = d / (b + v)

For the return journey upstream, the time taken is:

t_back = d / (b - v)

Setting Up the Equation

The total time for the round trip is the sum of the time taken for both journeys:

t = t_to + t_back

Substituting the expressions we derived:

t = d / (b + v) + d / (b - v)

Finding the Ratio

Now, we want to find the ratio of the time taken to go to the destination to the time taken to return:

Ratio = t_to / t_back

Using the expressions for t_to and t_back, we can express the ratio as:

Ratio = (d / (b + v)) / (d / (b - v))

The distance d cancels out, simplifying our ratio to:

Ratio = (b - v) / (b + v)

Final Thoughts

This ratio gives us a clear understanding of how the current affects the travel times in each direction. If the speed of the boat is significantly greater than the speed of the current, the ratio will be closer to 1, indicating that the times are more balanced. However, as the current speed approaches the boat's speed, the return journey will take considerably longer, reflecting a larger ratio.

In summary, the ratio of the time taken to go downstream to the time taken to return upstream is given by:

Ratio = (b - v) / (b + v)

This relationship highlights the impact of the river's current on the overall journey time, illustrating a fundamental concept in physics related to motion in varying mediums.