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In a closed system with no friction, a red sphere of 2.5 kg stands stationary. A blue sphere with a mass of 5.8 kg approaches the first sphere with a speed of 4.1 m/s. The two collide. After the collision, the blue sphere begins moving forward with a speed of 1.3 m/s. What is the velocity of the red sphere after the collision?
Red sphere 2.5 kg stationary velocity after ?
Blue sphere 5.8 kg speed 4.1 m/s after collision speed 1.3 m/s
m1 = the mass of the blue sphere = 5.8 kg
m2 = the mass of the red sphere = 2.5 kg
v1 = initial velocity of the blue sphere before the collision = 4.1 m/s
v2 = initial velocity of the red sphere before the collision = 0 m/s
v'1 = final velocity of the blue sphere after the collision = 1.3 m/s
v'2 = final velocity of the red sphere after the collision = ?
Conservation of momentum
m1v1 + m2v2 = m1v1 + m2v2
(5.8)(4.1) + (2.5)(0) = (5.8)(1.3) +(2.5)(v2)
23.78 = 7.54 + (2.5)(v2)
Subtract 7.54 from both sides
16.24 = (2.5)v2
2.5 2.5
v2 = 6.5
To find the velocity of the red sphere after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity (p = mv).
Before the collision:
The momentum of the red sphere is zero since it is stationary (m_red * v_red = 0).
The momentum of the blue sphere is given by the product of its mass (m_blue) and initial velocity (v_blue): p_blue_before = m_blue * v_blue.
After the collision:
The momentum of the red sphere, which is now moving, is given by the product of its mass (m_red) and velocity after the collision (v_red).
The momentum of the blue sphere, which is moving forward, is given by the product of its mass (m_blue) and velocity after the collision (v_blue_after).
Since the total momentum before the collision is equal to the total momentum after the collision, we can write the equation as:
m_blue * v_blue = m_red * v_red + m_blue * v_blue_after
Now we can plug in the given values:
m_red = 2.5 kg
m_blue = 5.8 kg
v_blue = 4.1 m/s
v_blue_after = 1.3 m/s
Substituting these values into the equation, we have:
5.8 kg * 4.1 m/s = 2.5 kg * v_red + 5.8 kg * 1.3 m/s
Now, solve for v_red:
23.78 kg·m/s = 2.5 kg * v_red + 7.54 kg·m/s
Subtracting 7.54 kg·m/s from both sides:
23.78 kg·m/s - 7.54 kg·m/s = 2.5 kg * v_red
16.24 kg·m/s = 2.5 kg * v_red
Dividing both sides by 2.5 kg:
v_red = 16.24 kg·m/s / 2.5 kg
v_red ≈ 6.496 m/s
Therefore, after the collision, the red sphere has a velocity of approximately 6.496 m/s.