Problem 41 - Introductory Chemistry Atoms First

Question

Given a speed of \(600.0\) miles/h and a measured time of \(50.0 \mathrm{~h}\), multiply the conversion factor by the measured time to make identical units cancel. Calculate the result and include its units.

Step-by-Step Solution

Answer

The total distance covered is \(30,000.0\) miles.

Step 1: Write down given information

We are given the following information: Speed = 600.0 mph Time = 50.0 hours

Step 2: Convert the speed to miles per hour (if necessary)

The speed is already given in miles per hour (mph), so we can skip this step.

Step 3: Calculate the distance covered using speed and time

To find the distance covered, we use the formula Distance = Speed × Time. Given: Speed = 600.0 mph Time = 50.0 hours Distance = 600.0 mph × 50.0 hours Distance = 30000.0 miles The total distance covered is 30,000.0 miles.

Key Concepts

Speed and Velocity

When it comes to understanding motion in physics, the concepts of speed and velocity are fundamental. Speed is defined as the distance traveled per unit of time and is a scalar quantity, which means it has magnitude but does not have a direction. The formula to calculate speed is given by:
\( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).
On the other hand, velocity is a vector quantity, which implies that it has both magnitude and direction. For instance, '60 miles per hour north' indicates a velocity because it mentions a direction. The difference between speed and velocity is particularly important in physics because it can influence how we calculate and understand motion.

For example, if a car travels around a circular track and ends up at the starting point after a certain time, its average speed is the total distance traveled divided by the elapsed time, while its average velocity is zero because there has been no displacement from the original position. This distinction is crucial for analyzing motion and solving problems related to transportation, astrophysics, or even sports.

Units of Measurement

Units of measurement are the standard quantities used to express physical quantities. In physics, having standardized units is essential for accuracy, clarity, and communication of measurements. For distance, units such as meters (m), kilometers (km), and miles (mi) are commonly used. Time is typically measured in seconds (s), minutes (min), or hours (h).

In the provided exercise, the speed is given in miles per hour (mph), which is a compound unit combining distance in miles and time in hours. Working within the same units or converting to compatible units is vital to solve physics problems correctly. For example, if speed were given in meters per second (m/s) but the distance needed to be found in kilometers, one would have to convert from meters to kilometers by understanding that 1000 meters make up 1 kilometer.

Using proper units of measurement is critical because incorrect units can lead to errors in calculations and interpretations. It's also important when communicating results, as standard units allow scientists and engineers around the world to understand and replicate experiments or calculations.

Distance-Time Relationship

The relationship between distance and time is one of the most fundamental concepts in kinematics, the branch of physics that deals with motion. The distance an object travels over a period of time depends on its speed or velocity. By knowing two of these variables, we can calculate the third. The basic formula connecting these three quantities is:
\( \text{Distance} = \text{Speed} \times \text{Time} \).

In our exercise, to find the distance, we multiply the speed of 600.0 mph by the time of 50.0 hours to obtain a distance of 30,000.0 miles. This calculation assumes a constant speed throughout the given time, which is often an idealization or approximation as most objects do not maintain a constant speed due to various factors such as acceleration or deceleration.

Graphically, the distance-time relationship can be represented by a distance-time graph where the slope of the line indicates the speed. A steeper slope means higher speed. Understanding this relationship not only helps in textbook exercises but is also essential for planning trips, analyzing traffic patterns, and in fields like aerospace engineering and astronomy.