The gravitational attraction between two objects with masses m_A and m_B, separated by distance 𝓍, is F = Gm_Am_B/𝓍², where G is the gravitational constant. If one mass is much greater than the other, the larger mass stays essentially at rest while the smaller mass moves toward it. Suppose a 1.5 x 10¹³ kg comet is passing the orbit of Mars, heading straight for the sun at a speed of 3.5 x 10⁴ m/s. What will its speed be when it crosses the orbit of Mercury? Astronomical data are given in the tables at the back of the book, and G = 6.67 x 10^-11 Nm²/kg².

Verified step by step guidance

1. Step 1: Identify the key variables and constants in the problem. The mass of the comet is m = 1.5 × 10¹³ kg, its initial speed is v₁ = 3.5 × 10⁴ m/s, and the gravitational constant is G = 6.67 × 10⁻¹¹ Nm²/kg². The problem involves the gravitational potential energy and kinetic energy of the comet as it moves closer to the Sun.
2. Step 2: Use the principle of conservation of energy. The total mechanical energy (kinetic energy + gravitational potential energy) of the comet remains constant as it moves toward the Sun. Write the equation for conservation of energy: $E_1=E_2$ where $E_1=K_1+U_1$ and $E_2=K_2+U_2$.
3. Step 3: Express the kinetic energy and gravitational potential energy at both points. Kinetic energy is given by $K=12m{v}^2$, and gravitational potential energy is given by $U=-GmMr$, where M is the mass of the Sun and r is the distance from the Sun. Substitute the values for the comet's initial and final positions (Mars and Mercury orbits) into these equations.
4. Step 4: Rearrange the conservation of energy equation to solve for the final speed v₂ of the comet when it crosses Mercury's orbit. The equation becomes: $K_1+U_1=K_2+U_2$. Substitute the expressions for kinetic and potential energy, and isolate v₂.
5. Step 5: Plug in the numerical values for the distances (Mars and Mercury orbits), the mass of the Sun, and the comet's mass into the equation. Perform algebraic manipulations to simplify the equation and solve for v₂. This will give the comet's speed when it crosses Mercury's orbit.

Key Concepts

Gravitational Force

The gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It states that the force (F) is proportional to the product of the two masses (mᴀ and mᴃ) and inversely proportional to the square of the distance (𝓍) between their centers. This relationship is quantified by the gravitational constant (G), which has a value of approximately 6.67 x 10⁻¹¹ Nm²/kg².

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant over time. In the context of gravitational interactions, as an object moves in a gravitational field, its potential energy decreases while its kinetic energy increases, ensuring that the total mechanical energy (kinetic + potential) remains constant. This principle is crucial for calculating the speed of the comet as it approaches the sun.

Kinetic and Potential Energy

Kinetic energy is the energy of an object due to its motion, calculated as KE = 1/2 mv², where m is mass and v is velocity. Potential energy in a gravitational field is given by PE = -G(m₁m₂)/r, where m₁ and m₂ are the masses and r is the distance between them. As the comet approaches the sun, its potential energy decreases while its kinetic energy increases, leading to a change in speed that can be calculated using these energy concepts.