Final answer:
The gravitational potential energy between two 5.00-kg spherical steel balls separated by 15.0 cm is -1.11 × 10-8 J. By conserving energy and assuming they begin at rest, each ball will reach an approximate speed of 1.49 × 10-5 m/s upon impact in deep space.
Step-by-step explanation:
Evaluation of Gravitational Potential Energy and Impact Speed
To evaluate the gravitational potential energy (GPE) between two 5.00-kg spherical steel balls separated by 15.0 cm, we use the formula for GPE between two masses:
GPE = -G x (m1 x m2) / r
Where G is the gravitational constant (6.674 × 10-11 N m2/kg2), m1 and m2 are the masses of the two objects, and r is the separation between their centers. Substituting in the given values:
GPE = -(6.674 × 10-11) x (5.00 x 5.00) / 0.15
GPE = -1.11 × 10-8 J
For part (b), assuming both spheres start at rest in deep space, we apply the conservation of energy to determine impact speed. As the spheres attract each other, gravitational potential energy is converted into kinetic energy. The initial potential energy will be equal to the combined kinetic energy of both spheres at impact.
Using the conservation of energy:
K1 + K2 = -GPE
Since both spheres have the same mass and start from rest, they will have the same speed upon impact. Hence, K1 = K2, and the total kinetic energy (K) at impact is K = 2 x (1/2) x m x v2.
K = -GPE
m x v2 = -GPE
v2 = -GPE / m
v = sqrt(-GPE / m)
v = sqrt((-1.11 × 10-8 J) / 5.00 kg)
v \u2248 1.49 × 10-5 m/s
Therefore, each sphere will be traveling at an approximate speed of 1.49 × 10-5 m/s when they collide.