Problem 80 - University Physics with Modern Physics

Question

Two students are canoeing on a river. While heading upstream, they accidentally drop an empty bottle overboard. They then continue paddling for 60 minutes, reaching a point 2.0 km farther upstream. At this point they realize that the bottle is missing and, driven by ecological awareness, they turn around and head downstream. They catch up with and retrieve the bottle (which has been moving along with the current) 5.0 km downstream from the turnaround point. (a) Assuming a constant paddling effort throughout, how fast is the river flowing? (b) What would the canoe speed in a still lake be for the same paddling effort?

Step-by-Step Solution

Answer

River speed: 3.0 km/h; Canoe speed in still water: 4.5 km/h.

Step 1: Define Variables

Let \( V_c \) be the speed of the canoe in still water, \( V_r \) be the speed of the river flow, and \( d \) be the distance traveled by the canoe upstream in 60 minutes, 2.0 km in total. Consequently, the speed against the current is \( V_c - V_r \). The speed with the current is \( V_c + V_r \).

Step 2: Determine Time Upstream and Downstream

The time to travel 2.0 km upstream is \( t_1 = \frac{2.0 \text{ km}}{V_c - V_r} \). After 60 minutes, or 1 hour, they turn around. The bottle has drifted 3.0 km downstream from the dropping point in 1 hour, given that they retrieve it 5.0 km downstream from the turnaround point.

Step 3: Analyze Bottle's Drift and Downstream Travel

Since the bottle drifted 3.0 km in 1 hour, we can deduce \( V_r = 3.0 \text{ km/h} \). Knowing that upon turning around and traveling 5.0 km downstream they retrieved the bottle, we can set up the equation for downstream speed: \( 2\cdot V_r = 6.0 \text{ km} \) with \( t_2 \) for their travel upstream and downstream: \( 2t_1 = \frac{6.0 \text{ km}}{V_c + 3 \text{ km/h}} \).

Step 4: Establish Canoe Speed in Still Water

Substitute \( V_r = 3.0 \text{ km/h} \) in the equation from Step 3: \[ \frac{2.0}{V_c - 3} + \frac{5.0}{V_c + 3} = 2 \]Solving this gives the speed of the canoe in still water.

Step 5: Solve the Equations

Simplify and solve \[ \frac{2.0}{V_c - 3} + \frac{5.0}{V_c + 3} = 2 \].Start by finding a common denominator and simplifying:\[ 2(V_c + 3) + 5(V_c - 3) = 2(V_c^2 - 9) \]\[ 2V_c + 6 + 5V_c - 15 = 2V_c^2 - 18 \]Combine and simplify:\[ 7V_c - 9 = 2V_c^2 - 18 \]\[ 0 = 2V_c^2 - 7V_c - 9 \]Now, solve this quadratic equation using the quadratic formula.

Step 6: Quadratic Formula Solution

Utilizing the quadratic formula: \( V_c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -7 \), \( c = -9 \), we find that \[ V_c = \frac{7 \pm \sqrt{49 + 72}}{4} \]Calculating the positive root, since speed cannot be negative: \[ V_c = \frac{7 + 11}{4} = \frac{18}{4} = 4.5 \text{ km/h} \].

Step 7: Conclusion of Speeds

With \( V_r = 3.0 \text{ km/h} \) and \( V_c = 4.5 \text{ km/h} \), we conclude that the river flows at 3.0 km/h, and the canoe speed in still water is 4.5 km/h.

Key Concepts

Upstream and Downstream Motion

When dealing with the flow of a river, understanding the concepts of upstream and downstream motion becomes crucial, especially in problems involving relative motions like canoeing. Imagine a river as a conveyor belt that can either assist or hinder movement depending on the direction you are heading.

• **Upstream Motion**: Here, you are moving against the current. For example, if the canoe has a speed of \( V_c \) in still water, the effective speed upstream is \( V_c - V_r \), where \( V_r \) is the speed of the river.
• **Downstream Motion**: This is when you're moving with the current. In this case, the effective speed of the canoe is \( V_c + V_r \). This is due to the assistance from the river's flow.

Knowing these basics simplifies calculating actual travel times and distances when the water's motion plays a role.

Paddling Effort

The paddling effort is the constant rate at which the canoeists paddle, regardless of direction. Their effort determines the speed of the canoe in still water. It is crucial to distinguish this from the actual speed in relation to the riverbank, as this depends on the current's assistance or resistance.

Even if the students maintain the same paddling effort throughout, as they did in our original exercise, their actual speed changes depending on the current.

* For example, when headed upstream, the river's resistance decreases their speed, but it doesn't mean the students are paddling with less effort.
* Conversely, downstream, their speed increases because the current aids their motion.

Relative Velocity

Relative velocity refers to the velocity of an object as observed from a particular frame of reference.The concept becomes handy when considering two moving objects like a canoe and the river in which it's paddling. Relative velocity allows us to understand the effective speed or velocity of the canoe in relation to the river's motion.

In a scenario like the canoeists', the speeds can be calculated as follows:

• **Canoe's velocity against the river (upstream)**: \( V_c - V_r \) is the effective speed.
• **Canoe's velocity with the river (downstream)**: \( V_c + V_r \) is the effective speed.

Understanding this concept is vital when calculating times and distances in problems involving moving mediums like rivers.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, usually in the format \( ax^2 + bx + c = 0 \). They play a significant role in solving problems related to motion, distances, speeds, and times when variables follow a squared relationship.

In the canoe problem, the quadratic equation arises from solving for the canoe's speed in still water. The relationship between upstream and downstream motions, thanks to paddling effort and river speed, can lead to a quadratic equation. Solving it involves:

• Identifying constants \( a \) (associated with \( x^2 \)), \( b \) (the linear coefficient), and \( c \) (the constant term).
• Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find possible values of \( V_c \), ensuring the speeds make sense (positive velocity).

These steps are critical in finding values that align with real-world constraints, like the speed of a canoe not being negative.