two stars with equal mass of 4.0x10^15 kg are separated by a distance of 2 km. They are moving in circular obits around the middle between the two stars. What is the speed of the stars?

To find the speed of the stars, we can use the concept of gravitational force and centripetal force.

1. Recall that the gravitational force between two objects is given by the formula:

F = G * (m₁ * m₂) / r²

Where F is the gravitational force, G is the gravitational constant (approximately 6.67430 × 10⁻¹¹ N m²/kg²), m₁ and m₂ are the masses of the two objects, and r is the distance between them.

2. In this case, we have two stars with equal masses (m₁ = m₂ = 4.0 × 10¹⁵ kg) and a distance between them (r) of 2 km. However, we need to convert the distance to meters since the gravitational constant is given in SI units. 1 km equals 1000 meters, so 2 km is equal to 2000 meters.

3. Now, we can calculate the gravitational force between the stars:

F = (6.67430 × 10⁻¹¹ N m²/kg²) * ((4.0 × 10¹⁵ kg)²) / (2000 m)²

Solving this equation will give us the value of the gravitational force.

4. Next, we need to realize that the gravitational force is responsible for providing the centripetal force required for circular motion. In circular motion, the centripetal force is given by:

F = m * v² / r

Where F is the centripetal force, m is the mass of the object, v is its velocity, and r is the radius of the circular path (in this case, the distance between the stars).

5. Since we have two stars with equal mass, we can use the mass of one star (4.0 × 10¹⁵ kg) in the formula.

F = (4.0 × 10¹⁵ kg) * v² / (2000 m)

6. Now, equate the gravitational force (from step 3) to the centripetal force (from step 5) and solve the equation for v (velocity).

(6.67430 × 10⁻¹¹ N m²/kg²) * ((4.0 × 10¹⁵ kg)²) / (2000 m)² = (4.0 × 10¹⁵ kg) * v² / (2000 m)

7. Rearrange the equation to solve for v:

v² = [(6.67430 × 10⁻¹¹ N m²/kg²) * ((4.0 × 10¹⁵ kg)²)] / [(4.0 × 10¹⁵ kg) * 2000 m]

v² = (6.67430 × 10⁻¹¹ N m²/kg²) * (4.0 × 10¹⁵ kg) / (2000 m)

Calculate the value on the right-hand side of the equation.

8. Finally, take the square root of v² to get the speed (v) of the stars.

This process will help you find the speed of the stars. Make sure to plug in the correct values and units to get an accurate result.