A car sits on an entrance ramp to a freeway, waiting for a break in the traffic. Then the driver accelerates with constant acceleration along the ramp and onto the freeway. The car starts from rest, moves in a straight line, and has a speed of 20 m/s(45 mi/h) when it reaches the end of the 120-m-long ramp. (b) How much time does it take the car to travel the length of the ramp?

Verified step by step guidance

1. Identify the given values: initial velocity (v_i) = 0 m/s (since the car starts from rest), final velocity (v_f) = 20 m/s, and displacement (s) = 120 m.
2. Use the kinematic equation that relates initial velocity, final velocity, acceleration, and displacement: v_f^2 = v_i^2 + 2as. Solve this equation for acceleration (a).
3. Rearrange the equation to isolate acceleration: a = (v_f^2 - v_i^2) / (2s).
4. Use the kinematic equation that relates initial velocity, final velocity, acceleration, and time: v_f = v_i + at. Solve this equation for time (t).
5. Rearrange the equation to isolate time: t = (v_f - v_i) / a. Substitute the value of acceleration found in step 3 to find the time.

Key Concepts

Constant Acceleration

Constant acceleration refers to a situation where an object's velocity changes at a steady rate over time. In this case, the car accelerates uniformly from rest, meaning its speed increases consistently until it reaches 20 m/s. This concept is crucial for applying kinematic equations to determine the time taken to travel a specific distance.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. For this problem, the equation that connects distance, initial velocity, acceleration, and time will be used to find the time it takes for the car to travel the 120 meters of the ramp.

Displacement

Displacement is the distance an object moves in a specific direction. In this scenario, the car's displacement is the length of the ramp, which is 120 meters. Understanding displacement is essential for solving the problem, as it provides the distance over which the car accelerates and helps in applying the kinematic equations effectively.