A 0.395-kg blue bead slides on a frictionless, curved wire, starting from rest at point in the figure below, where h = 1.50 m. At point , the blue bead collides elastically with a 0.650-kg green bead at rest. Find the maximum height the green bead rises as it moves up the wire.

Well, it seems we have a colorful situation here! A blue bead and a green bead getting into some elastic fun on a curved wire.

To find the maximum height the green bead rises, we need to determine the conservation of energy during the collision.

Since the collision is elastic, the total kinetic energy before the collision should be equal to the total kinetic energy after the collision.

The initial kinetic energy of the blue bead is determined by its mass (0.395 kg) and its velocity (which we'll call v). Since it starts from rest, its initial kinetic energy is zero.

The initial kinetic energy of the green bead is also zero since it is at rest.

After the collision, the blue bead and the green bead will exchange velocities. Let's call the final velocity of the blue bead Vb, and the final velocity of the green bead Vg.

Using the conservation of energy, we can write:
0.5 * m1 * v^2 + 0.5 * m2 * 0^2 = 0.5 * m1 * Vb^2 + 0.5 * m2 * Vg^2

Plugging in the values:
0.5 * 0.395 * v^2 + 0.5 * 0.650 * 0^2 = 0.5 * 0.395 * Vb^2 + 0.5 * 0.650 * Vg^2

Since the blue bead continues downwards after the collision, Vb will be negative.

Now, we need to figure out the relationship between Vb and Vg. Since the collision is elastic, the relative magnitude of the velocities should be the same before and after the collision:

|m1 * v| = |m1 * Vb| = |m2 * Vg|

Substituting the values:
0.395 * v = 0.395 * (-Vb) = 0.650 * Vg

From this equation, we can solve for Vg in terms of v:
Vg = (0.395 * v) / 0.650

Now, to find the maximum height the green bead rises, we can use the conservation of mechanical energy.

The total mechanical energy at the top of the curved wire is equal to the sum of the gravitational potential energy and the kinetic energy at that point.

0.5 * m2 * Vg^2 + m2 * g * h = 0.5 * m2 * 0^2 + m2 * g * H

Since the initial kinetic energy of the green bead is zero, the equation simplifies to:
m2 * g * h = m2 * g * H

Canceling out the mass and the acceleration due to gravity:
h = H

Therefore, the maximum height the green bead rises is equal to the initial height h, which is 1.50 m.

So, the green bead goes up to the same height it started from! Perhaps it saw something interesting up there and decided to take a second look.

To find the maximum height the green bead rises as it moves up the wire after the collision with the blue bead, we can apply the principle of conservation of mechanical energy.

According to the principle of conservation of mechanical energy, the initial mechanical energy of the system (blue bead + green bead) is equal to the final mechanical energy of the system.

The initial mechanical energy of the system is given by the potential energy of the blue bead at point A:

E_initial = m_blue * g * h

where:
m_blue = mass of the blue bead = 0.395 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height of point A = 1.50 m

The final mechanical energy of the system is given by the potential energy of the green bead at its maximum height:

E_final = m_green * g * h_rise

where:
m_green = mass of the green bead = 0.650 kg
g = acceleration due to gravity = 9.8 m/s^2
h_rise = maximum height the green bead rises

Since the collision between the blue bead and the green bead is elastic, the total mechanical energy of the system is conserved. Therefore, we have:

E_initial = E_final

m_blue * g * h = m_green * g * h_rise

Substituting the given values:

0.395 kg * 9.8 m/s^2 * 1.50 m = 0.650 kg * 9.8 m/s^2 * h_rise

Simplifying the equation:

5.786 J = 6.37 h_rise

Dividing both sides of the equation by 6.37:

h_rise = 5.786 J / 6.37 kg = 0.908 m

Therefore, the maximum height the green bead rises as it moves up the wire after the collision is 0.908 meters.

To find the maximum height the green bead rises, we can apply the principle of conservation of mechanical energy. This principle states that the total mechanical energy of a system remains constant if no external forces (such as friction) are acting on it.

In this case, before the collision, the blue bead has some potential energy at point A due to its height h. After the collision, the blue bead transfers some of its kinetic energy to the green bead. As a result, the green bead will gain kinetic energy, converting it into potential energy as it rises up the wire.

Let's break down the problem step by step:

1. Calculate the initial potential energy of the blue bead at point A.
- The equation for potential energy is given by PE = m * g * h, where m is the mass of the bead, g is the acceleration due to gravity, and h is the height.
- Plug in the values: m = 0.395 kg, g = 9.8 m/s^2, h = 1.50 m.
- Calculate the initial potential energy: PE = 0.395 kg * 9.8 m/s^2 * 1.50 m.

2. Determine the initial kinetic energy of the blue bead before the collision.
- Since the blue bead starts from rest, its initial velocity is 0.
- The equation for kinetic energy is given by KE = 0.5 * m * v^2, where m is the mass of the bead and v is its velocity.
- Plug in the values: m = 0.395 kg, v = 0.
- Calculate the initial kinetic energy: KE = 0.5 * 0.395 kg * (0 m/s)^2.

3. Calculate the total mechanical energy before the collision.
- The total mechanical energy is the sum of the initial potential energy and the initial kinetic energy: total energy = PE + KE.

4. Determine the final kinetic energy of the blue and green beads after the collision.
- Since the collision is elastic, both beads conserve kinetic energy. The total kinetic energy before and after the collision remains the same.
- We need to find the final velocity of the blue bead and the green bead.
- The equation for kinetic energy is KE = 0.5 * m * v^2, where m is the mass of the bead and v is its velocity.
- Plug in the values for the blue bead: m = 0.395 kg, v = final velocity of the blue bead.
- Plug in the values for the green bead: m = 0.650 kg, v = final velocity of the green bead.
- After the collision, the green bead starts moving and the blue bead continues its motion but with a changed velocity.

5. Determine the final potential energy of the system when the green bead reaches its maximum height.
- As the green bead rises, its kinetic energy is converted into potential energy.
- At the maximum height, the green bead's velocity will be zero.
- The equation for potential energy is given by PE = m * g * h, where m is the mass of the bead, g is the acceleration due to gravity, and h is the height.
- Plug in the values for the green bead: m = 0.650 kg, g = 9.8 m/s^2, h = maximum height of the green bead.

6. Calculate the maximum height of the green bead.
- The maximum height is the final potential energy of the system when the green bead reaches its maximum height.
- Plug in the values and calculate the maximum height: PE = 0.650 kg * 9.8 m/s^2 * h.

By following these steps, you should be able to find the maximum height the green bead rises as it moves up the wire.

m1=0.395 kg, h=1.5 m, m2= 0.65 kg, H=?

m1•g•h=m1•v ²/2
v =sqrt(2•g•h)=sqrt(2•9.8•1.5)=5.4 m/s
m1•v = m1•v1+m2•v2
For the case of elastic collision
v2=2 •m1•v/(m1+m2)= =2•0.395•5.4/(0.395+0.65) =2.67 m/s
m2•v2 ²/2= m2•g•H
H= v2 ²/2•g=2.67²/2•9.8=0.36 m