Problem 59 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A NASA satellite has just observed an asteroid that is on a collision course with the Earth. The asteroid has an estimated mass, based on its size, of \(5 \times 10^{9} \mathrm{~kg}\). It is approaching the Earth on a head-on course with a velocity of \(660 \mathrm{~m} / \mathrm{s}\) relative to the Earth and is now \(5.0 \times 10^{6} \mathrm{~km}\) away. With what speed will it hit the Earth's surface, neglecting friction with the atmosphere?

Step-by-Step Solution

Answer

The asteroid will hit the Earth's surface with a speed of approximately 11180 m/s.

Step 1: Convert Distance to Meters

First, convert the distance from kilometers to meters. The asteroid is currently \(5.0 \times 10^{6} \text{ km} = 5.0 \times 10^9 \text{ m}\).

Step 2: Use Conservation of Energy

Use the conservation of energy to determine the final velocity of the asteroid. Initially, the asteroid has kinetic energy \(KE_i = \frac{1}{2}mv_i^2\) and gravitational potential energy \(U_i = -\frac{G M e m}{r_0}\), where \(M_e\) is the Earth's mass, \(r_0\) is the initial distance, and \(G\) is the gravitational constant.

Step 3: Express Initial Energy

Calculate the initial total energy: \[ E_i = \frac{1}{2} m v_i^2 - \frac{G M_e m}{r_0} \] where \(m = 5 \times 10^9 \text{ kg}\), \(v_i = 660 \text{ m/s}\), \(G = 6.674 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}\), \(M_e = 5.972 \times 10^{24} \text{ kg}\), and \(r_0 = 5.0 \times 10^9 \text{ m}\).

Step 4: Express Final Energy

For final energy when the asteroid hits the Earth's surface: \[ E_f = \frac{1}{2} m v_f^2 - \frac{G M_e m}{R_e} \] where \(v_f\) is the final velocity and \(R_e = 6.371 \times 10^6 \text{ m}\) is the Earth's radius.

Step 5: Solve Conservation Equation

Since energy is conserved \(E_i = E_f\) hence, \[ \frac{1}{2} m v_i^2 - \frac{G M_e m}{r_0} = \frac{1}{2} m v_f^2 - \frac{G M_e m}{R_e} \]Cancel mass \(m\) and solve for \(v_f\).

Step 6: Calculate Final Velocity

Rearrange to find: \[ v_f = \sqrt{v_i^2 + 2 G M_e \left( \frac{1}{R_e} - \frac{1}{r_0} \right)} \]Substitute the values to find \(v_f \approx 11180 \text{ m/s}\).

Key Concepts

Conservation of Energy

In physics, the conservation of energy principle states that energy cannot be created or destroyed, although it can change forms. This concept is crucial in understanding how asteroids interact with Earth.
Consider an asteroid on a collision course with Earth. It possesses both kinetic and gravitational potential energy. As the asteroid gets closer, these energy types transform, but the total energy remains constant.

• Initial total energy = sum of initial kinetic energy and initial gravitational potential energy.
• Final total energy = sum of final kinetic energy and final gravitational potential energy at Earth's surface.

By ensuring the initial energy equals the final energy, we can predict the asteroid's speed when it hits Earth. This conversion and conservation are crucial in calculating real-world scenarios like asteroid impacts.

Gravitational Potential Energy

Gravitational potential energy (U) is the energy an object possesses because of its position in a gravitational field. For objects far from Earth or any massive body, it is defined as\[ U = -\frac{G M_em}{r} \]where:

• \(G\) is the gravitational constant.
• \(M_e\) is Earth's mass.
• \(m\) is the mass of the object, in this case, the asteroid.
• \(r\) is the distance from the object to the center of the Earth.

In this exercise, the asteroid's potential energy changes as it moves from its initial distance toward Earth's center. As the asteroid approaches Earth, it loses gravitational potential energy, as \(r\) decreases, transforming it into kinetic energy. This conversion is key to calculating the final speed when the asteroid impacts Earth's surface.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is given by the formula\[ KE = \frac{1}{2}mv^2 \]where:

• \(m\) is the mass of the object, which, for an asteroid, can be quite large.
• \(v\) is the speed of the object.

In the context of an asteroid, as it travels towards Earth, its speed increases, causing its kinetic energy to grow. Initially, the asteroid has a certain kinetic energy based on its initial speed and mass. As it descends into Earth's gravitational field, kinetic energy increases due to the conversion from gravitational potential energy. Calculating the increase in kinetic energy allows us to determine the asteroid's final speed just before impact.