A small and a large sphere, of mass M and 3M respectively, are arranged as shown on the left side of the figure above. The spheres are then simultaneously dropped from rest. When the large sphere strikes the floor, the spheres have fallen a height H. Assume air resistance is negligible. Immediately after striking the floor, the large sphere is moving upward with speed vb and collides head-on with the small sphere, which is moving downward with identical speed vb at that instant. Immediately after the collision, the small sphere moves upward with speed vS and the large sphere has speed vL.
QUESTION: Please Derive an equation that relates vb, vS, and vL.
In this particular situation vL = 0.
this link will help you get your answer. there's no space between h and ttp
h ttp://dev.physicslab.org/Document.aspx?doctype=1&filename=questions_2012B2.xml
Well, well, well, looks like we have some physics to clown around with!
Let's start by considering the conservation of momentum before and after the collision. Since the momentum is conserved in a closed system, we can write:
mL * vb + mS * (-vb) = mL * vL + mS * vS,
where mL and mS are the masses of the large and small spheres, respectively.
Now we can substitute the values given: mL = 3M and mS = M. Also, since vL = 0, we can simplify the equation to:
3M * vb - M * vb = M * vS.
Combining like terms, we get:
2M * vb = M * vS.
Dividing both sides by M, we obtain:
2vb = vS.
So, the relationship that relates vb, vS, and vL is:
2vb = vS.
Hope that helps, and remember, physics can be a real circus sometimes!
To derive an equation that relates vb, vS, and vL, we can consider the conservation of momentum and the conservation of kinetic energy during the collision.
1. Conservation of momentum:
Before the collision, the momentum of the system is given by the sum of the momenta of the large and small spheres:
Momentum before collision = (mass of large sphere) * vb - (mass of small sphere) * vb
= 3M * vb - M * vb
= 2M * vb
After the collision, the momentum of the system is given by the sum of the momenta of the large and small spheres:
Momentum after collision = (mass of large sphere) * vL - (mass of small sphere) * vS
Since the large sphere initially has no velocity (vL = 0), the equation becomes:
2M * vb = - M * vS
2. Conservation of kinetic energy:
Before the collision, the kinetic energy of the system is given by the sum of the kinetic energies of the large and small spheres:
Kinetic energy before collision = (1/2) * (mass of large sphere) * vb^2 + (1/2) * (mass of small sphere) * vb^2
= (1/2) * 3M * vb^2 + (1/2) * M * vb^2
= (5/2) * M * vb^2
After the collision, the kinetic energy of the system is given by the sum of the kinetic energies of the large and small spheres:
Kinetic energy after collision = (1/2) * (mass of large sphere) * vL^2 + (1/2) * (mass of small sphere) * vS^2
Since the large sphere initially has no velocity (vL = 0), the equation becomes:
(5/2) * M * vb^2 = (1/2) * M * vS^2
Now we have two equations:
2M * vb = - M * vS
(5/2) * M * vb^2 = (1/2) * M * vS^2
We can simplify these equations by canceling out the common factors of M:
2 * vb = - vS
(5/2) * vb^2 = (1/2) * vS^2
Finally, we have derived the equation that relates vb, vS, and vL in this specific situation where vL = 0:
2 * vb = - vS
(5/2) * vb^2 = (1/2) * vS^2
To derive an equation that relates vb, vS, and vL, we can use the principle of conservation of momentum and the principle of conservation of kinetic energy.
1. Conservation of momentum:
Since the large sphere is initially at rest and the small sphere is moving downward with speed vb, the total momentum before the collision is given by:
Initial momentum = 3M * 0 + M * vb
After the collision, the small sphere moves upward with speed vS, and the large sphere has speed vL. The total momentum after the collision is given by:
Final momentum = M * vS + 3M * (-vL)
According to the principle of conservation of momentum, the initial momentum is equal to the final momentum:
3M * 0 + M * vb = M * vS + 3M * (-vL)
Simplifying the equation, we have:
M * vb = M * vS - 3M * vL
2. Conservation of kinetic energy:
Since only gravitational potential energy is involved, the total kinetic energy is conserved. Initially, the spheres have gravitational potential energy only, and after the collision, they have both kinetic energy and gravitational potential energy.
Before the collision, the total kinetic energy is zero:
Initial kinetic energy = 0
After the collision, the total kinetic energy is given by:
Final kinetic energy = (1/2) * M * vS^2 + (1/2) * 3M * vL^2
According to the principle of conservation of kinetic energy, the initial and final kinetic energies should be equal:
0 = (1/2) * M * vS^2 + (1/2) * 3M * vL^2
Simplifying the equation, we have:
M * vS^2 + 3M * vL^2 = 0
Combining the equation from the conservation of momentum with the equation from the conservation of kinetic energy, we have the following system of equations:
M * vb = M * vS - 3M * vL
M * vS^2 + 3M * vL^2 = 0
Given that vL = 0, we can substitute vL = 0 in the first equation:
M * vb = M * vS - 3M * 0
M * vb = M * vS
Finally, canceling out the common factor of M on both sides of the equation, we get:
vb = vS
So, the equation that relates vb, vS, and vL is vb = vS.