Final answer:
The question involves finding the velocity of a billiard ball after an elastic collision, using principles of conservation of momentum and energy in physics. By applying these principles to the provided velocities and directions, one can calculate the final velocity of the second ball.
Step-by-step explanation:
To solve this problem, we can use the principle of conservation of linear momentum. According to this principle, the total linear momentum of an isolated system remains constant before and after a collision or explosion.
The linear momentum (p) is given by the product of mass (m) and velocity (v):
p=m⋅v
The total initial linear momentum (pinitial ) of the system is the sum of the individual momenta of the three parts:
pinitial = mobject ⋅ vobject + m1⋅ v1 + m2 . v2
where
mobject is the mass of the object,
vobject is the velocity of the object,
m1 is the mass of the first part,
v1 is the velocity of the first part,
m2 is the mass of the second part,
v2 is the velocity of the second part.
Since the explosion breaks the object into three parts, the third part's mass and velocity can be expressed in terms of the total mass and velocity of the object and the masses and velocities of the first two parts.
After the explosion, the total final linear momentum (pfinal) is the sum of the individual momenta of the three parts:
pfinal = m1 ⋅ v1f + m2 ⋅ v2f + m3 ⋅ v3f
where v1f , v2f , and v3f are the final velocities of the first, second, and third parts, respectively.
Since linear momentum is conserved, pinitial = pfinal.
Now, let's use these equations to solve the problem:
Given data:
mobject = 23.0kg
vobject = 28.0m/s
m1 = 5.80kg
v1 = 56.0m/s
First, calculate the initial linear momentum:
pinitial = mobject ⋅ vobject + m1 ⋅ v1
pinitial = (23.0kg⋅28.0m/s)+(5.80kg⋅56.0m/s)
pinitial = 644.0kg⋅m/s+324.8kg⋅m/s
pinitial = 968.8kg⋅m/s
Now, we know that pinitial = pfinal, so:
pfinal = 968.8kg⋅m/s
Now, let's express the final linear momentum in terms of the masses and velocities of the three parts. Let m2 be the mass of the second part, and
v2 be its velocity. The third part will have a mass of m3 =mobject −m1−m2 , and its velocity will be the negative of the sum of the velocities of the object, the first part, and the second part, since the explosion causes the third part to move in the opposite direction. So:
pfinal = m1 ⋅v1f+m2⋅v2f+m3 ⋅v3f
968.8kg⋅m/s=(5.80kg⋅v1f)+(m2⋅v2f)+((23.0kg−5.80kg−m2)⋅(−vobject−v1−v2))
Now, you can solve for m2, v2f , and v3f. The solution will depend on the specific values of these variables, and you may need additional information or assumptions to fully determine them.