A sprinter can accelerate with constant acceleration for 4.0 s before reaching top speed. He can run the 100 meter dash in 10.0 s. What is his speed as he crosses the finish line?

Verified step by step guidance

1. Identify the two phases of the sprinter's motion: (1) constant acceleration for 4.0 s and (2) constant velocity for the remaining time. Use kinematic equations to analyze each phase.
2. For the first phase, use the kinematic equation: $d=\frac{1}{2}a{t}^2$, where $d$ is the distance covered during acceleration, $a$ is the acceleration, and $t$ is the time (4.0 s).
3. At the end of the acceleration phase, calculate the sprinter's velocity using the equation: $v=at$, where $v$ is the velocity, $a$ is the acceleration, and $t$ is 4.0 s.
4. For the second phase, calculate the distance covered at constant velocity using the equation: $d=vt$, where $d$ is the distance, $v$ is the velocity from the first phase, and $t$ is the remaining time (10.0 s - 4.0 s).
5. Combine the distances from both phases to ensure the total distance is 100 m. Solve for the acceleration $a$ and use it to find the final velocity $v$ as the sprinter crosses the finish line.

Key Concepts

Constant Acceleration

Constant acceleration refers to a situation where an object's velocity changes at a steady rate over time. In the context of the sprinter, this means that his speed increases uniformly during the first 4.0 seconds of his run. The equations of motion for constant acceleration can be applied to determine the distance covered and the final speed at the end of the acceleration phase.

Kinematic Equations

Kinematic equations are mathematical formulas that describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. For the sprinter, these equations can be used to calculate the distance he covers while accelerating and to find his final speed as he crosses the finish line after the acceleration phase.

Average Speed

Average speed is defined as the total distance traveled divided by the total time taken. In this scenario, the sprinter's average speed over the entire 100 meters can be calculated by dividing 100 meters by 10 seconds. This average speed provides insight into his performance, but it is important to distinguish it from his instantaneous speed at the finish line, which may be higher due to the acceleration phase.