A pilot who accelerates at more than $4 g$ begins to 'gray out' but doesn't completely lose consciousness. Assuming constant acceleration, what is the shortest time that a jet pilot starting from rest can take to reach Mach $4$ (four times the speed of sound) without graying out? (Use $331$ m/s for the speed of sound in cold air.)

검증된 단계별 안내

First, understand the concept of 'g-force'. The term 'g' refers to the acceleration due to gravity, which is approximately 9.81 m/s². When a pilot accelerates at more than 4g, it means the acceleration is more than 4 times 9.81 m/s².

Calculate the maximum allowable acceleration without graying out: $ a_{max} = 4 \times 9.81 \text{ m/s}^2 $.

Determine the target speed, which is Mach 4. Since the speed of sound is given as 331 m/s, Mach 4 is $ v_{target} = 4 \times 331 \text{ m/s} $.

Use the kinematic equation for constant acceleration to find the shortest time to reach the target speed: $ v = a \cdot t $. Rearrange to solve for time: $ t = \frac{v_{target}}{a_{max}} $.

Substitute the values for $ v_{target} $ and $ a_{max} $ into the equation to find the shortest time: $ t = \frac{4 \times 331}{4 \times 9.81} $.

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주요 개념

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

Acceleration and g-force

Acceleration is the rate of change of velocity over time. In this context, 'g-force' refers to the acceleration experienced as a multiple of Earth's gravitational acceleration (9.8 m/s²). A pilot experiencing more than 4g can begin to gray out, meaning the acceleration should not exceed 4 times 9.8 m/s² to avoid this effect.

Mach number

The Mach number is a dimensionless unit representing the ratio of the speed of an object to the speed of sound in the surrounding medium. Mach 4 means the object is traveling at four times the speed of sound. In this problem, the speed of sound is given as 331 m/s, so Mach 4 corresponds to a speed of 1324 m/s.

Kinematic equations

Kinematic equations describe the motion of objects under constant acceleration. The relevant equation here is v = at, where v is the final velocity, a is the acceleration, and t is the time. Solving for time when acceleration and final velocity are known allows us to determine the shortest time to reach Mach 4 without exceeding 4g.