A boat moves from P to Q (downstream) in 2.5 h and from Q to P (upstream) in 5.0 h respectivel

step 1 Let speed of the river be speed of river is equal to u kilometer per hour and speed of the boat

A boat moves from P to Q (downstream) in 2.5 h and from Q...

A boat moves from $\mathrm{P}$ to $\mathrm{Q}$ (downstream) in $2.5 \mathrm{~h}$ and from $\mathrm{Q}$ to $\mathrm{P}$ (upstream) in $5.0 \mathrm{~h}$ respectively, when the engine is producing maximum speed on the boat in each case. If due to increased water flow, the speed of flow of the river gets doubled, how long will it take the boat (moving with maximum possible speed) to move from (i) P to Q (downstream)? (ii) Q to P (upstream)?

Key Concepts

Relative Motion in Fluids

This concept deals with how the actual or effective speed of a boat is determined by both its speed in still water (its engine’s maximum speed) and the speed of the river current. Downstream, the current aids the boat, increasing its effective speed by addition, while upstream, the current opposes the boat, reducing its effective speed by subtraction. This understanding is crucial in problems that involve navigating in flowing water, as the net speed directly affects travel times.

Distance-Speed-Time Relationship

At the core of the problem is the basic relation that time equals distance divided by speed. This principle allows one to compute travel times by knowing the distance covered and the effective speed of the boat (which is impacted by the river’s current). When the river flow changes, recalculating the effective speed and subsequently the time using this relationship is essential.

Impact of Variable Flow Conditions

This concept involves understanding how changes in the river's current, such as an increase or doubling of the flow speed, affect the boat's effective speed in both downstream and upstream conditions. Doubling the flow can significantly alter the net speed of the boat and thus the time needed to cover the same distance. Recognizing the interplay between the boat's own speed and the water current is vital for analyzing how modifications in environmental conditions affect travel times.

Transcript

Please give Ace some feedback