In 2000, NASA placed a satellite in orbit around an asteroid. Consider a spherical asteroid with a mass of 1.0 x 10¹⁶ kg and a radius of 8.8 km. (b) What is the escape speed from the asteroid?

Verified step by step guidance

1. Identify the mass (M) of the asteroid and the radius (R) from the problem statement. Here, M = 1.0 x 10^16 kg and R = 8.8 km (convert this to meters by multiplying by 1000, so R = 8800 m).
2. Recall the formula for escape speed, which is given by v_{escape} = \sqrt{\frac{2GM}{R}}, where G is the gravitational constant (6.674 x 10^{-11} m^3 kg^{-1} s^{-2}).
3. Substitute the values of G, M, and R into the escape speed formula.
4. Calculate the value inside the square root first, which is 2GM/R.
5. Finally, take the square root of the result from the previous step to find the escape speed.

Key Concepts

Escape Velocity

Escape velocity is the minimum speed an object must reach to break free from the gravitational pull of a celestial body without any additional propulsion. It depends on the mass of the body and the distance from its center to the point of escape. The formula for escape velocity (v) is given by v = √(2GM/r), where G is the gravitational constant, M is the mass of the body, and r is the radius from the center of the mass to the point of escape.

Gravitational Constant

The gravitational constant (G) is a fundamental physical constant that quantifies the strength of gravitational attraction between two masses. Its value is approximately 6.674 × 10⁻¹¹ N(m/kg)². This constant is crucial in calculations involving gravitational forces and escape velocities, as it allows us to relate mass and distance to the gravitational force experienced by an object.

Mass and Radius of Celestial Bodies

The mass and radius of a celestial body, such as an asteroid, are essential parameters in determining its gravitational influence. The mass affects the strength of the gravitational field, while the radius determines how far from the center of the mass the gravitational force acts. In the context of escape velocity, both parameters are used to calculate the speed required for an object to overcome the gravitational pull of the body.