A planet orbiting a distant star has radius 3.24 × 10⁶ m. The escape speed for an object launched from this planet’s surface is 7.65 × 10³ m/s. What is the acceleration due to gravity at the surface of the planet?

검증된 단계별 안내

Start by recalling the formula for escape speed, which is given by \( v_e = \sqrt{2gr} \), where \( v_e \) is the escape speed, \( g \) is the acceleration due to gravity, and \( r \) is the radius of the planet.

Rearrange the formula to solve for \( g \): \( g = \frac{v_e^2}{2r} \).

Substitute the given values into the equation: \( v_e = 7.65 \times 10^3 \text{ m/s} \) and \( r = 3.24 \times 10^6 \text{ m} \).

Calculate \( v_e^2 \) by squaring the escape speed: \( (7.65 \times 10^3)^2 \).

Divide the result of \( v_e^2 \) by \( 2r \) to find the acceleration due to gravity \( g \).

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주요 개념

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Escape Velocity

Escape velocity is the minimum speed an object must reach to break free from a celestial body's gravitational pull without further propulsion. It depends on the mass and radius of the body. For a planet, the escape velocity (v) is given by the formula v = sqrt(2 * G * M / R), where G is the gravitational constant, M is the planet's mass, and R is its radius.

Gravitational Acceleration

Gravitational acceleration (g) is the acceleration experienced by an object due to the gravitational force exerted by a massive body like a planet. It is calculated using the formula g = G * M / R^2, where G is the gravitational constant, M is the mass of the planet, and R is its radius. This acceleration determines the weight of objects on the planet's surface.

Relationship Between Escape Velocity and Gravitational Acceleration

The escape velocity and gravitational acceleration are related through the planet's mass and radius. By equating the escape velocity formula v = sqrt(2 * G * M / R) with the gravitational acceleration formula g = G * M / R^2, we can derive g = v^2 / (2 * R). This relationship allows us to calculate the gravitational acceleration if the escape velocity and radius are known.