Drag racer acceleration The fastest drag racers can reach a speed of 330 mi/hr over a quarter-mile strip in 4.45 seconds (from a standing start). Complete the following sentence about such a drag racer: At some point during the race, the maximum acceleration of the drag racer is at least _____ mi/hr/s.        .

Verified step by step guidance

1. First, understand that the problem involves finding the maximum acceleration of the drag racer. Acceleration is the rate of change of velocity with respect to time.
2. To find the maximum acceleration, we can use the Mean Value Theorem for derivatives, which states that for a continuous and differentiable function, there exists at least one point where the instantaneous rate of change (derivative) equals the average rate of change over the interval.
3. Calculate the average acceleration over the entire race. The average acceleration is given by the change in velocity divided by the time taken. Here, the change in velocity is from 0 to 330 mi/hr, and the time is 4.45 seconds.
4. Convert the average acceleration from mi/hr/s to a consistent unit if necessary, such as mi/hr^2 or ft/s^2, to make it easier to compare with instantaneous acceleration.
5. According to the Mean Value Theorem, the maximum acceleration at some point during the race must be at least equal to this average acceleration, since the velocity function is continuous and differentiable over the time interval.

Key Concepts

Acceleration

Acceleration is the rate of change of velocity over time. In this context, it measures how quickly the drag racer increases its speed from a standing start to its maximum speed. It is typically expressed in units such as miles per hour per second (mi/hr/s), indicating how many miles per hour the speed increases for each second of acceleration.

Average Acceleration

Average acceleration can be calculated by taking the change in velocity and dividing it by the time taken for that change. For the drag racer, this involves determining the difference between the final speed (330 mi/hr) and the initial speed (0 mi/hr), then dividing by the total time (4.45 seconds). This provides a straightforward way to estimate the minimum acceleration during the race.

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. It provides the equations and principles necessary to analyze the motion of the drag racer, including how distance, speed, and time relate to one another. Understanding kinematics is essential for solving problems related to acceleration and velocity in this context.