Question

Predict \& Explain You drive your car in a straight line at $15\mathrm{m}/\mathrm{s}$ for $10\mathrm{km}$, then at $25\mathrm{m}/\mathrm{s}$ for another $10\mathrm{km}$. (a) Is your average speed for the entire trip more than, less than, or equal to $20\mathrm{m}/\mathrm{s}$ ? (b) Choose the best explanation from the following: A. More time is spent driving at $15\mathrm{m}/\mathrm{s}$ than at $25\mathrm{m}/\mathrm{s}$. B. The average of $15\mathrm{m}/\mathrm{s}$ and $25\mathrm{m}/\mathrm{s}$ is $20\mathrm{m}/\mathrm{s}$. C. Less time is spent driving at $15\mathrm{m}/\mathrm{s}$ than at $25\mathrm{m}/\mathrm{s}$.

Short answer

Average speed is less than 20 m/s because more time is spent at 15 m/s.

Step-by-step solution

Calculate Time for Each Segment

To determine the time spent at each speed, use the formula $t=\frac{d}{v}$, where $t$ is time, $d$ is distance, and $v$ is speed. For the first segment at $15\mathrm{m}/\mathrm{s}$, the distance is $10\mathrm{km}=10,000\mathrm{m}$. Thus, the time is $t_1=\frac{10,000}{15}\approx 666.67\mathrm{s}$. For the second segment at $25\mathrm{m}/\mathrm{s}$, the time is $t_2=\frac{10,000}{25}=400\mathrm{s}$.

Key concepts

Uniform Motion

Uniform motion refers to motion where an object travels a straight line distance at a constant speed. This means both the direction and magnitude of the speed stay the same throughout the motion. In such cases, the relationship between distance traveled, time taken, and speed is straightforward because there is no change in speed or direction. For instance, in the given exercise, the car moves at a consistent speed for each segment of its journey. In the first segment, it travels at $15\mathrm{m}/\mathrm{s}$ and in the second segment at $25\mathrm{m}/\mathrm{s}$.
Both segments exhibit uniform motion, meaning the car doesn't accelerate or decelerate until a change in segments. This uniform speed allows us to use simple formulas to determine time and distance, leading to an effective analysis of the trip.
Understanding uniform motion simplifies problems where the total motion consists of different segments completed at constant speeds. Each segment can be considered separately and added together to establish the desired parameters of the journey, such as total time or average speed.

Distance and Time Relationship

The relationship between distance, time, and speed is a fundamental aspect of motion. In uniform motion, this relationship is expressed through the formula:

• $t=\frac{d}{v}$, where $t$ is time, $d$ is distance, and $v$ is speed.

This formula allows you to calculate how much time it takes to travel a specific distance at a given speed. For example, converting kilometers to meters for the first, $10\mathrm{km}=10,000\mathrm{m}$, helps apply this formula easily. With a speed of $15\mathrm{m}/\mathrm{s}$, you find:

• $t_1=\frac{10,000}{15}\approx 666.67\mathrm{s}$

For the second segment, at $25\mathrm{m}/\mathrm{s}$, calculation is straightforward as:

• $t_2=\frac{10,000}{25}=400\mathrm{s}$

Grasping the distance and time relationship in uniform motion allows for efficient speed analysis over different segments, enabling a comprehensive understanding of both individual segment and total journey outcomes.

Speed Conversions

Converting speed units is a useful skill for understanding motion across various measuring systems. In physics problems, speed is often needed in specific units, which require adjustments for accurate calculations. Typically, speed might be given in meters per second (m/s) when dealing with scientific computations. However, other practical situations might require kilometers per hour (km/h).
To convert from $\mathrm{m}/\mathrm{s}$ to $\mathrm{km}/\mathrm{h}$, use the conversion factor where $1\mathrm{m}/\mathrm{s}=3.6\mathrm{km}/\mathrm{h}$. For instance, given speeds in the problem:

• $15\mathrm{m}/\mathrm{s}$ converts to $15\times 3.6=54\mathrm{km}/\mathrm{h}$
• $25\mathrm{m}/\mathrm{s}$ converts to $25\times 3.6=90\mathrm{km}/\mathrm{h}$

Recognizing the need for speed conversions helps when you compare and analyze motion figures from different sources. This gives you an advantage when you encounter real-world scenarios requiring unit interpretation across diverse systems, ensuring consistency in calculations and discussions.