### Problem 4**Given:**- Sphere A: - Mass, \( m_a = 0.06 \) kg - Velocity magnitude, \( v_a = 3.0 \) m/s - Direction: making an angle \( \theta = 25^\circ \) with the \( +x \)-axis.- Sphere B: - Mass, \( m_b = 0.04 \) kg - Velocity magnitude, \( v_b = 4.0 \) m/s - Direction: along the \( +y \)-axis.- Sphere C: - Mass, \( m_c = 0.05 \) kg - Velocity magnitude, \( v_c = 5.0 \) m/s - Direction: along the \( +x \)-axis.**Scenario:**The spheres A, B, and C are all approaching the origin as they slide on a frictionless surface. They collide and stick together at the origin.**Questions to Solve:**(a) What is the final velocity of the combined object?(b) How does the final kinetic energy compare to the kinetic energy before the collision?### Explanation of the Problem:We need to find the final velocity of the combined mass of spheres after they stick together and also analyze the kinetic energy before and after the collision.### 1. **Final Velocity Calculation:**To find the final velocity of the combined object, we use the principle of conservation of momentum. The combined object's final momentum will be the vector sum of the initial momenta of all three spheres.**Components of the Initial Momentum:**- Sphere A: - \( p_{ax} = m_a \cdot v_a \cdot \cos(\theta) = 0.06 \cdot 3.0 \cdot \cos(25^\circ) \) - \( p_{ay} = m_a \cdot v_a \cdot \sin(\theta) = 0.06 \cdot 3.0 \cdot \sin(25^\circ) \)- Sphere B: - \( p_{bx} = 0 \) - \( p_{by} = m_b \cdot v_b = 0.04 \cdot 4.0 \)- Sphere C: - \( p_{cx} = m_c \cdot v_c

Name: ### Problem 4**Given:**- Sphere A: - Mass, \( m_a = 0.06 \) kg - Ve
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Description: ### Problem 4**Given:**- Sphere A: - Mass, \( m_a = 0.06 \) kg - Velocity magnitude, \( v_a = 3.0 \) m/s - Direction: making an angle \( \theta = 25^\circ \) with the \( +x \)-axis.- Sphere B: - Mass, \( m_b = 0.04 \) kg - Velocity magnitude, \

ma = 0.06 kg va = 3 m/s θa = 25omb = 0.04 kg vb = 4 m/s along +y axis mc = 0.05 kg vc = 5 m/s along…

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4. Sphere A has mass ma = 0.06 kg, velocity magnitude va - 3.0 m/s making = an angle 25° with the + axis. Sphere B has mass m = 0.04 kg, velocity magnitude Ub = 4.0 m/s and moves along the +y axis. Sphere C has mass mc = 0.05 kg, velocity magnitude vc = 5.0 m/s and moves along the +x axis. See figure. They are all approaching the origin as they slide on a frictionless surface. The three spheres arrive at the origin at the same time and stick together. (a) What is the final velocity of the combined object? (b) How does the final kinetic energy compare to the kinetic energy before the collision?

4. Sphere A has mass ma = 0.06 kg, velocity magnitude va - 3.0 m/s making = an angle 25° with the + axis. Sphere B has mass m = 0.04 kg, velocity magnitude Ub = 4.0 m/s and moves along the +y axis. Sphere C has mass mc = 0.05 kg, velocity magnitude vc = 5.0 m/s and moves along the +x axis. See figure. They are all approaching the origin as they slide on a frictionless surface. The three spheres arrive at the origin at the same time and stick together. (a) What is the final velocity of the combined object? (b) How does the final kinetic energy compare to the kinetic energy before the collision?

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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### Problem 4

**Given:**

- Sphere A:
- Mass, \( m_a = 0.06 \) kg
- Velocity magnitude, \( v_a = 3.0 \) m/s
- Direction: making an angle \( \theta = 25^\circ \) with the \( +x \)-axis.

- Sphere B:
- Mass, \( m_b = 0.04 \) kg
- Velocity magnitude, \( v_b = 4.0 \) m/s
- Direction: along the \( +y \)-axis.

- Sphere C:
- Mass, \( m_c = 0.05 \) kg
- Velocity magnitude, \( v_c = 5.0 \) m/s
- Direction: along the \( +x \)-axis.

**Scenario:**

The spheres A, B, and C are all approaching the origin as they slide on a frictionless surface. They collide and stick together at the origin.

**Questions to Solve:**

(a) What is the final velocity of the combined object?
(b) How does the final kinetic energy compare to the kinetic energy before the collision?

### Explanation of the Problem:

We need to find the final velocity of the combined mass of spheres after they stick together and also analyze the kinetic energy before and after the collision.

### 1. **Final Velocity Calculation:**

To find the final velocity of the combined object, we use the principle of conservation of momentum.

The combined object's final momentum will be the vector sum of the initial momenta of all three spheres.

**Components of the Initial Momentum:**

- Sphere A:
- \( p_{ax} = m_a \cdot v_a \cdot \cos(\theta) = 0.06 \cdot 3.0 \cdot \cos(25^\circ) \)
- \( p_{ay} = m_a \cdot v_a \cdot \sin(\theta) = 0.06 \cdot 3.0 \cdot \sin(25^\circ) \)

- Sphere B:
- \( p_{bx} = 0 \)
- \( p_{by} = m_b \cdot v_b = 0.04 \cdot 4.0 \)

- Sphere C:
- \( p_{cx} = m_c \cdot v_c

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