Question

A canoeist paddled upstream at 10 meters per minute, turned around, and drifted downstream at 15 meters per minute. If the distance traveled in each direction was the same, and the time spent turning the canoe around was negligible, what was the canoeist's average speed over the course of the journey, in meters per minute? A 11.5 B 12 C 12.5 D 13 E 13.5

Short answer

B 12

Step-by-step solution

Define Variables

Let the distance traveled in each direction be denoted as $d$ meters and the times spent upstream and downstream be $t_u$ and $t_d$ respectively.

Set Up Equations for Time

Time can be calculated by dividing distance by speed. So, $t_u=\frac{d}{10}$ (upstream) and $t_d=\frac{d}{15}$ (downstream).

Find Total Time

The total time for the entire journey is the sum of the times spent upstream and downstream: $t_{total}=t_u+t_d=\frac{d}{10}+\frac{d}{15}$.

Combine Time Fractions

Find a common denominator to combine the fractions: $t_{total}=\frac{3d}{30}+\frac{2d}{30}=\frac{5d}{30}=\frac{d}{6}$.

Calculate Average Speed

Average speed is the total distance divided by total time. The total distance traveled is $2d$ (going and returning): $v_{avg}=\frac{2d}{\frac{d}{6}}=2d\times \frac{6}{d}=12$.

Choose the Correct Option

The average speed of the canoeist over the whole journey is 12 meters per minute, which corresponds to option B.

Key concepts

Distance Calculation

In this exercise, the key is to understand the concept of distance calculation. Distance is how far an object has traveled in a straight line. Here, the canoeist paddles upstream and downstream, covering the same distance in both directions. Distance in each direction is denoted by the variable $d$. This helps simplify calculations using one variable. For total distance, you sum up the distance for both upstream and downstream: total distance = $d+d=2d$. It's crucial always to start with a clear understanding of what's being calculated to solve the problem correctly.

Speed and Velocity

Speed and velocity might sound similar but are distinct. Speed is how fast an object moves, while velocity includes direction. In this case, the canoeist's speeds are different for upstream and downstream. Upstream paddle speed = 10 meters per minute and downstream drift speed = 15 meters per minute. These speeds are crucial for calculating time.
Time taken for each leg of the journey is given by:$$t_u=\frac{d}{\text{speed upstream}}=\frac{d}{10}$$
$$t_d=\frac{d}{\text{speed downstream}}=\frac{d}{15}$$
Always separate different speeds and times before finding the total or average speed. This organizes the steps clearly and makes solving easier.

Time Management

Understanding how to manage and calculate time is key in this exercise. Total travel time is the sum of the times for upstream and downstream: $t_{total}=t_u+t_d$. Simplify using a common denominator: $$t_{total}=\frac{d}{10}+\frac{d}{15}=\frac{3d}{30}+\frac{2d}{30}=\frac{5d}{30}=\frac{d}{6}$$
With total time in hand, you can determine average speed. Average speed is found by dividing total distance by total time: $$v_{avg}=\frac{2d}{\frac{d}{6}}=2d\times \frac{6}{d}=12\text{ meters per minute}$$ Always breaking down the total time calculation helps in efficiently finding the solution.