A $15$ -kg rock is dropped from rest on the earth and reaches the ground in $1.75$ s. When it is dropped from the same height on Saturn's satellite Enceladus, the rock reaches the ground in $18.6$ s. What is the acceleration due to gravity on Enceladus?

검증된 단계별 안내

Identify the known values for the problem on Earth: the mass of the rock (15 kg), the time it takes to reach the ground (1.75 s), and the acceleration due to gravity on Earth (9.81 m/s²).

Use the kinematic equation for free fall on Earth to find the height from which the rock is dropped: $ h = \frac{1}{2} g t^2 $, where $ g = 9.81 \text{ m/s}^2 $ and $ t = 1.75 \text{ s} $.

Calculate the height $ h $ using the values for Earth. This height will be the same for the drop on Enceladus.

Use the same kinematic equation for the drop on Enceladus: $ h = \frac{1}{2} g_{Enceladus} t^2 $, where $ t = 18.6 \text{ s} $ and $ h $ is the height calculated from the Earth scenario.

Solve for the acceleration due to gravity on Enceladus $ g_{Enceladus} $ by rearranging the equation: $ g_{Enceladus} = \frac{2h}{t^2} $. Substitute the known values to find $ g_{Enceladus} $.

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주요 개념

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

Acceleration due to Gravity

Acceleration due to gravity is the rate at which an object speeds up as it falls freely under the influence of gravity. On Earth, this is approximately 9.81 m/s². It varies depending on the celestial body, affecting how quickly objects fall.

Free Fall Motion

Free fall motion describes the movement of an object solely under the influence of gravity, without any air resistance. The time it takes for an object to reach the ground can be used to calculate the gravitational acceleration using kinematic equations.

Kinematic Equations

Kinematic equations relate the variables of motion: displacement, initial velocity, final velocity, acceleration, and time. For an object in free fall, these equations can determine the acceleration due to gravity by using the time of fall and initial conditions.