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. How much time does it take the car to travel the length of the ramp?

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Identify the known values: initial velocity $ v_0 = 0 $ m/s (since the car starts from rest), final velocity $ v = 20 $ m/s, and displacement $ s = 120 $ m.

Use the kinematic equation that relates velocity, acceleration, and displacement: $ v^2 = v_0^2 + 2as $. Since $ v_0 = 0 $, this simplifies to $ v^2 = 2as $.

Rearrange the equation to solve for acceleration $ a $: $ a = \frac{v^2}{2s} $. Substitute the known values to find $ a $.

Use the kinematic equation $ v = v_0 + at $ to solve for time $ t $. Since $ v_0 = 0 $, it simplifies to $ t = \frac{v}{a} $.

Substitute the values of $ v $ and $ a $ into the equation $ t = \frac{v}{a} $ to find the time it takes for the car to travel the length of the ramp.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Constant Acceleration

Constant acceleration refers to a situation where an object's velocity changes at a steady rate over time. In this problem, the car accelerates uniformly, meaning its acceleration remains the same throughout its motion along the ramp. This allows us to use kinematic equations to determine the time taken to travel a specific distance.

Kinematic Equations

Kinematic equations are mathematical formulas used to describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. For this problem, the equation v^2 = u^2 + 2as can be used, where v is final velocity, u is initial velocity, a is acceleration, and s is displacement, to find the time taken.

Initial Conditions

Initial conditions are the starting values of variables in a physics problem, such as initial velocity and position. Here, the car starts from rest, meaning its initial velocity is zero. This simplifies calculations, as the initial velocity term in kinematic equations becomes zero, allowing us to focus on the relationship between acceleration, final velocity, and displacement.