Question

After two hours, a car is traveling at a velocity of 50 mph. After another hour, the same car is traveling at a velocity of -25 mph. If the velocity of the car is continuous, is there any given time between two and three hours when the car stops? Justify your reasoning.

Option 1: Yes, the car stops at some point between two and three hours.
Option 2: No, the car never stops between two and three hours.
Option 3: The car stops exactly at the three-hour mark.
Option 4: The car stops exactly at the two-hour mark.

Answer 1

Yes, the car stops at some point between two and three hours because the velocity changes from 50 mph to -25 mph over that interval, implying there must be an instant where the velocity is 0 mph as the direction of motion changes.

Step-by-step explanation:

Option 1: Yes, the car stops at some point between two and three hours. To understand this, we need to consider the concept of velocity and how it changes over time. Velocity can be thought of as speed with a direction. Here we have a car traveling at 50 mph at the two-hour mark and then having a velocity of -25 mph at the three-hour mark. The negative sign indicates that the direction of the car's velocity has reversed. Since the velocity is continuous, by the Intermediate Value Theorem, there must be a point between two and three hours where the velocity of the car is 0 mph, meaning the car stops.

Given that velocity vs. time can be graphed and the value of velocity changes from positive to negative, it intersects the time-axis indicating that the car's velocity is zero, hence the car stops. The moment when the car's velocity is zero is when the car transitions from moving forward to moving backward, at that instant the car must come to a stop before reversing its direction.