Problem 17 - Thomas Calculus

Question

The speedometer on your car reads a steady 35 mph. Could you be accelerating? Explain.

Step-by-Step Solution

Answer

Yes, you could be accelerating if the car changes direction while maintaining constant speed.

Step 1: Understand Speed vs. Velocity

Speed is a measure of how fast an object is moving, while velocity is speed with a direction. A constant speed, such as 35 mph, indicates how fast your car is moving regardless of the direction.

Step 2: Define Acceleration Concept

Acceleration occurs when there is a change in velocity, which means any change in the speed or direction of an object. So, even if speed remains constant, acceleration can occur if the direction changes.

Step 3: Analyze the Speedometer Reading

A speedometer displays only the magnitude of speed, not direction. Hence, it doesn't measure changes in direction, and a steady speed of 35 mph only reflects constant speed.

Step 4: Consider Possible Acceleration

Although the speedometer shows a constant reading, if the direction of the car is changing, such as in taking a turn, the velocity is changing, since velocity includes direction. Thus, the car could be accelerating.

Step 5: Conclusion Based on Definitions

Given that acceleration involves changes in velocity (either speed or direction), and since the car might be changing direction while maintaining constant speed, it's possible for the car to be accelerating.

Key Concepts

Understanding Speed vs. Velocity

Speed and velocity are foundational concepts in physics that relate to how fast an object is moving. Speed is a scalar quantity that refers to how fast something moves, without regard to its direction. For instance, driving at 35 mph tells you how quickly your car is covering ground.

Velocity, on the other hand, is a vector. This means it takes direction into account. When you say your car is moving at a velocity of 35 mph north, you are specifying both the speed and the direction. Thus, while your speed tells you how fast, your velocity tells you where you are going as well.

• Speed: Just the amount of ground covered over time.
• Velocity: The speed of the object in a specific direction.

Understanding this difference is crucial, especially when examining situations involving acceleration.

What Does Constant Speed Imply?

When a car maintains a constant speed, such as a steady 35 mph, it indicates that the rate of motion along the road remains unchanged over time. The car's speedometer reports this constant speed but doesn't offer insights into any directional changes.

Maintaining a constant speed doesn't mean that there are no changes in the car's motion. It merely indicates that the car's speed remains the same. However, even at a constant speed, the car can still experience motion changes.

• Constant Speed: The numerical value displayed by the speedometer remains unvaried.
• It does not imply any directional data.

This distinction becomes important when considering the possibility of the car accelerating.

The Impact of Direction Change on Acceleration

Acceleration isn't only about speeding up. It also occurs when there is a change in direction. This is because velocity changes any time the speed or direction of an object varies.

Imagine a car moving at a steady speed of 35 mph. If it takes a turn, the direction of the moving car changes while maintaining constant speed. This effectively alters its velocity and creates acceleration, despite the unchanged speed.

• Change in Direction: Alters velocity.
• Even at constant speed, these direction changes indicate acceleration.

Acceleration then, can occur in situations where the speed is steady but the travel direction shifts. Recognizing the role of direction is key to fully understanding vehicle acceleration.