Boats and Streams Concepts

1. Upstream & Downstream

What is the question? Find the speed of a boat in still water or the speed of the stream when upstream/downstream speeds or distances and times are given.

Formula:

$$\text{Downstream speed}=u+v$$ $$\text{Upstream speed}=u-v$$

where:

• $u$ = speed of boat in still water
• $v$ = speed of stream

Therefore:

$$u=\frac{D+U}{2}$$ $$v=\frac{D-U}{2}$$

Example: A boat travels 48 km downstream in 4 hours and upstream in 6 hours. Find the speed of the boat in still water and the speed of the stream.

Solution:

Downstream speed:

$$D=\frac{48}{4}=12\text{ km/h}$$

Upstream speed:

$$U=\frac{48}{6}=8\text{ km/h}$$

Speed in still water:

$$u=\frac{12+8}{2}=10\text{ km/h}$$

Stream speed:

$$v=\frac{12-8}{2}=2\text{ km/h}$$

Answer:

Boat speed = 10 km/h
Stream speed = 2 km/h

---

2. Meeting of Boats in a River

What is the question? Two boats start from different places and move towards each other. Find when or where they meet.

Formula:

When two boats move towards each other:

$$\text{Relative speed}=v_1+v_2$$ $$\text{Time}=\frac{\text{Distance}}{\text{Relative speed}}$$

If both speeds are given in still water, the stream cancels when they move towards each other.

Example: Two boats have speeds of 6 km/h and 10 km/h in still water. They start from opposite banks 48 km apart. The stream speed is 2 km/h. Find the time taken to meet.

Solution:

Since they move towards each other, the stream affects one boat downstream and the other upstream:

Boat A → 6 + 2 = 8 km/h

Boat B → 10 - 2 = 8 km/h

Combined speed:

$$8+8=16\text{ km/h}$$

Time:

$$t=\frac{48}{16}=3\text{ hours}$$

Answer:

3 hours

---

3. River Crossing and Downstream Drift

What is the question? A boat crosses a river while the current pushes it downstream. Find the crossing time, downstream drift, or distance travelled.

Formula:

If the boat is directed straight across the river:

$$\text{Crossing time} ==================== \frac{\text{River width}}{\text{Boat speed}}$$ $$\text{Drift} ============ \text{Stream speed}\times\text{Crossing time}$$

Therefore:

$$\text{Drift} ============ \frac{\text{River width}\times\text{Stream speed}} {\text{Boat speed}}$$

Example: A river is 100 m wide and flows at 3 m/s. A boat moves straight across the river at 5 m/s. Find the downstream drift.

Solution:

Crossing time:

$$t=\frac{100}{5}=20\text{ s}$$

Drift:

$$3\times20=60\text{ m}$$

Answer:

Drift = 60 m downstream

---

4. Floating Object / Dead Body / Log

What is the question? An object floats with the river. Find the distance it travels or the time taken to reach a particular point.

Formula:

A freely floating object moves with the stream:

$$\text{Object speed}=\text{Stream speed}$$

And:

$$D=St$$

Example: A bottle falls into a river flowing at 4 km/h. How far will it drift in 45 minutes?

Solution:

Convert 45 minutes into hours:

$$45\text{ min}=\frac{45}{60}=0.75\text{ h}$$

Distance:

$$D=4\times0.75=3\text{ km}$$

Answer:

3 km

---

5. Boat Goes Upstream and Returns Downstream

What is the question? A boat travels a certain distance upstream and comes back downstream. Find the total time, average speed, or distance.

Formula:

$$t_{\text{up}}=\frac{D}{u-v}$$ $$t_{\text{down}}=\frac{D}{u+v}$$

Total time:

$$T=\frac{D}{u-v}+\frac{D}{u+v}$$

For equal distances, average speed is:

$$\text{Average speed} ==================== \frac{2(u+v)(u-v)} {(u+v)+(u-v)}$$

which simplifies to:

$$\text{Average speed} ==================== \frac{u^2-v^2}{u}$$

Example: A boat travels 30 km upstream and returns 30 km downstream. Its speed in still water is 10 km/h and the stream speed is 2 km/h. Find the total time.

Solution:

Upstream speed:

$$10-2=8\text{ km/h}$$

Downstream speed:

$$10+2=12\text{ km/h}$$

Upstream time:

$$\frac{30}{8}=3.75\text{ h}$$

Downstream time:

$$\frac{30}{12}=2.5\text{ h}$$

Total:

$$3.75+2.5=6.25\text{ h}$$

Answer:

6.25 hours
= 6 hours 15 minutes

---

6. Find Distance from Upstream and Downstream Times

What is the question? The boat’s speed and stream speed are known, but the time taken upstream/downstream is given. Find the distance.

Formula:

$$D=(u-v)t_{\text{up}}$$

or

$$D=(u+v)t_{\text{down}}$$

Example: A boat moves upstream at 6 km/h and takes 5 hours. Find the distance travelled.

Solution:

$$D=6\times5$$ $$D=30\text{ km}$$

Answer:

30 km

---

7. Find Boat/Stream Speed from Upstream and Downstream Speeds

What is the question? The upstream and downstream speeds are given directly. Find the speed of the boat in still water and the speed of the stream.

Formula:

$$u=\frac{D+U}{2}$$ $$v=\frac{D-U}{2}$$

Example: A boat travels downstream at 18 km/h and upstream at 10 km/h. Find its speed in still water and the speed of the stream.

Solution:

$$u=\frac{18+10}{2}=14$$ $$v=\frac{18-10}{2}=4$$

Answer:

Boat speed = 14 km/h
Stream speed = 4 km/h

---

8. Boat Travels the Same Distance Upstream and Downstream

What is the question? A boat covers the same distance upstream and downstream. Compare the times taken or find one speed when the other is known.

Formula:

For the same distance:

$$\frac{t_{\text{up}}}{t_{\text{down}}} ===================================== \frac{u+v}{u-v}$$

Example: A boat’s speed in still water is 12 km/h and the stream speed is 4 km/h. Find the ratio of upstream time to downstream time for the same distance.

Solution:

Upstream speed:

$$12-4=8$$

Downstream speed:

$$12+4=16$$

For the same distance, time is inversely proportional to speed:

t_{\text{up}}:t_{\text{down}} ============================= # 16:8 2:1

Answer:

Upstream time : Downstream time = 2 : 1

---

9. Shortest-Time River Crossing

What is the question? A boat needs to cross a river in the minimum possible time. Find the time taken or the direction of travel.

Formula:

For minimum crossing time, the boat is directed perpendicular to the river bank.

$$t=\frac{W}{u}$$

where:

• $W$ = width of river
• $u$ = boat speed in still water

The stream only causes downstream drift.

Example: A river is 200 m wide. A boat can travel at 5 m/s in still water. Find the minimum time needed to cross.

Solution:

$$t=\frac{200}{5}=40\text{ s}$$

Answer:

40 seconds

---

10. Reaching the Point Directly Opposite

What is the question? A boat must reach the point directly opposite its starting point, without being carried downstream. Find the required direction or effective crossing speed.

Formula:

The boat must aim upstream so that its upstream component cancels the stream:

$$u\sin\theta=v$$

The effective speed across the river is:

$$u_{\text{across}} ================= \sqrt{u^2-v^2}$$

Therefore:

$$t= \frac{W}{\sqrt{u^2-v^2}}$$

Example: A river flows at 3 m/s. A boat can travel at 5 m/s in still water. What is its effective speed directly across the river?

Solution:

$$u_{\text{across}} ================= \sqrt{5^2-3^2}$$ $$=\sqrt{25-9}$$ $$=4\text{ m/s}$$

Answer:

Effective crossing speed = 4 m/s

---

11. Relative Speed of Boats in a River

What is the question? Two boats move in the same or opposite directions in a river. Find how quickly the distance between them changes.

Formula:

Same direction:

$$\text{Relative speed}=|v_1-v_2|$$

Opposite directions:

$$\text{Relative speed}=v_1+v_2$$

Example: Two boats move downstream at 12 km/h and 8 km/h. They are 20 km apart. How long will the faster boat take to catch the slower boat?

Solution:

Relative speed:

$$12-8=4\text{ km/h}$$

Time:

$$t=\frac{20}{4}=5\text{ hours}$$

Answer:

5 hours