A falling stone takes 0.28 s to travel past a window 2.2 m tall (Fig. 2–49). From what height above the top of the window did the stone fall?

검증된 단계별 안내

Identify the known quantities: The time it takes for the stone to pass the window is 0.28 s, and the height of the window is 2.2 m. Assume the stone is in free fall, so the acceleration due to gravity is \( g = 9.8 \; \text{m/s}^2 \).

Determine the velocity of the stone as it passes the top of the window. Use the kinematic equation for constant acceleration: \( h = v_0 t + \frac{1}{2} a t^2 \). Here, \( h = 2.2 \; \text{m} \), \( t = 0.28 \; \text{s} \), and \( a = g = 9.8 \; \text{m/s}^2 \). Solve for the initial velocity \( v_0 \) at the top of the window.

Once \( v_0 \) is determined, use it to calculate the height from which the stone fell above the window. The stone's velocity at the top of the window is related to its initial height by the kinematic equation: \( v_0^2 = v_i^2 + 2 a h \), where \( v_i = 0 \; \text{m/s} \) (since the stone starts from rest), \( a = g \), and \( h \) is the height above the window.

Rearrange the equation \( v_0^2 = 2 g h \) to solve for \( h \): \( h = \frac{v_0^2}{2 g} \). Substitute the value of \( v_0 \) obtained in the previous step and \( g = 9.8 \; \text{m/s}^2 \) to find the height.

Add the height of the window (2.2 m) to the calculated height above the window to determine the total height from which the stone fell.

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주요 개념

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

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, acceleration, and time. In this problem, kinematic equations will be used to relate the distance the stone falls to the time it takes to pass the window.

Acceleration due to Gravity

Acceleration due to gravity is the rate at which an object accelerates towards the Earth when in free fall, typically denoted as 'g' and approximately equal to 9.81 m/s². This constant is crucial for calculating the distance fallen by the stone, as it influences how quickly the stone accelerates as it descends.

Free Fall

Free fall refers to the motion of an object under the influence of gravity alone, with no other forces acting on it, such as air resistance. In this scenario, the stone is in free fall as it descends past the window, allowing us to apply the equations of motion to determine the height from which it fell.