Ball bearings are made by letting spherical drops of molten metal fall inside a tall tower—called a shot tower—and solidify as they fall. If a bearing needs 4.0 s to solidify enough for impact, how high must the tower be?

검증된 단계별 안내

Identify the key variables in the problem: the time of free fall is given as 4.0 seconds, and we need to find the height of the tower. Assume the acceleration due to gravity, \( g \), is \( 9.8 \; \text{m/s}^2 \), and the initial velocity \( v_0 \) is 0 since the drop starts from rest.

Use the kinematic equation for vertical motion under constant acceleration: \( h = v_0 t + \frac{1}{2} g t^2 \). Since \( v_0 = 0 \), the equation simplifies to \( h = \frac{1}{2} g t^2 \).

Substitute the known values into the simplified equation: \( h = \frac{1}{2} (9.8) (4.0)^2 \).

Simplify the expression inside the parentheses: \( h = \frac{1}{2} (9.8) (16.0) \).

Multiply the terms to find the height \( h \). This will give the required height of the tower in meters.

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주요 개념

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

Free Fall

Free fall refers to the motion of an object under the influence of gravity alone, without any air resistance. In this scenario, the molten metal drops from a height and accelerates downward due to gravitational force, which is approximately 9.81 m/s² on Earth. Understanding free fall is essential to calculate the height of the tower based on the time it takes for the metal to solidify.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. For free-falling objects, one key equation relates distance, initial velocity, time, and acceleration: d = v₀t + 0.5at². In this case, the initial velocity (v₀) is zero, allowing us to simplify the equation to d = 0.5gt², which is crucial for determining the height of the shot tower.

Solidification Time

Solidification time is the duration required for a material to transition from a liquid to a solid state. In this context, the molten metal needs 4.0 seconds to solidify sufficiently before impacting the ground. This time frame directly influences the calculation of the height of the tower, as it determines how long the metal will be in free fall before reaching the bottom.