Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a car's speedometer is constant, then the car cannot be accelerating.

Short answer

The statement is false. A car with a constant speedometer reading can still be accelerating if it's changing direction, as acceleration considers both magnitude (speed) and direction of velocity. An example of this is a car moving in a circular path at constant speed.

Step-by-step solution

Analyze the Statement

The statement in question is 'If a car's speedometer is constant, then the car cannot be accelerating.' A speedometer measures speed - the magnitude part of the velocity. To assess the truth of this statement, one needs to understand the nature of acceleration. Acceleration occurs when there is a change in velocity. Now, velocity is a vector quantity, meaning it has both magnitude (speed) and direction. So even if the magnitude of velocity (speed) is constant, there can still be acceleration if there is change in direction.

Formulate an Example

Let's consider a car moving in a circular path at a constant speed. Even though the speed is constant (the speedometer reading doesn't change), the direction in which the car is moving is continuously changing. Since acceleration is a vector and depends on changes in either speed or direction (or both), the car in this case is actually accelerating due to the continuous change in direction, even though the speed is constant. Consequently, the original statement is false.

Explain the Example

Since the car on a circular path is constantly changing direction, it experiences centripetal acceleration directed towards the center of the circle, even though its speed remains constant. This invalidates the original statement that a constant speed (as per a speedometer) means there can be no acceleration.