Conservation Of Momentum In One Dimension Tutorial Homework Answers

Understanding conservation ofmomentum in one dimension is fundamental for solving countless physics problems, especially those involving collisions and explosions. This tutorial provides clear, step-by-step guidance and solutions to common homework problems, helping you master the core principle that the total momentum of an isolated system remains constant. By the end, you'll be equipped to confidently tackle any one-dimensional momentum conservation problem.

Introduction Momentum, defined as the product of an object's mass and velocity (p = m*v), is a vector quantity, meaning it has both magnitude and direction. Conservation of momentum states that within an isolated system (where no external forces act), the total momentum before any interaction (like a collision) equals the total momentum after the interaction. This principle, derived from Newton's laws, is incredibly powerful for predicting outcomes in physics scenarios. This tutorial focuses specifically on solving homework problems involving one-dimensional motion, where all objects move along a straight line. We'll cover the essential steps, provide worked examples, explain the underlying physics, and address common questions.

Steps for Solving Conservation of Momentum Problems (One Dimension)

1. Identify the System: Clearly define the objects involved in the interaction (the system). Ensure the system is isolated (no significant external forces acting on it during the interaction).
2. Determine Initial Velocities (p_initial): Measure or calculate the momentum of each object before the interaction. Remember momentum is a vector, so direction is crucial. Assign a positive direction (e.g., right = positive, left = negative).
3. Determine Final Velocities (p_final): Measure or calculate the momentum of each object after the interaction. Again, assign the same positive direction.
4. Apply Conservation of Momentum: Write the fundamental equation: Total Initial Momentum = Total Final Momentum. Mathematically: Σp_initial = Σp_final.
5. Solve for the Unknown: Rearrange the equation to solve for the unknown quantity (e.g., velocity, mass, direction). Perform the necessary calculations.
6. Check Units and Sign: Ensure your answer has the correct units (kg·m/s). Verify the sign (+/-) matches the direction you assigned.

Worked Examples (Step-by-Step Solutions)

• Example 1: Elastic Collision (Objects Stick Together - Perfectly Inelastic)

  • Problem: A 2.0 kg toy car moving at 3.0 m/s collides head-on with a stationary 1.5 kg toy truck. After the collision, they stick together. What is their combined velocity?
  • Solution:
    • System: Car + Truck.
    • Initial Momentum (Car): p_car_initial = m_car * v_car_initial = 2.0 kg * 3.0 m/s = 6.0 kg·m/s (right, +).
    • Initial Momentum (Truck): p_truck_initial = 1.5 kg * 0 m/s = 0 kg·m/s (stationary).
    • Total Initial Momentum: Σp_initial = 6.0 kg·m/s + 0 kg·m/s = 6.0 kg·m/s.
    • Final State: Combined mass = 2.0 kg + 1.5 kg = 3.5 kg. Let final velocity be v_f (direction unknown yet).
    • Total Final Momentum: Σp_final = (3.5 kg) * v_f.
    • Conservation: Σp_initial = Σp_final => 6.0 kg·m/s = 3.5 kg * v_f.
    • Solve: v_f = 6.0 kg·m/s / 3.5 kg = 1.71 m/s (approximately).
    • Answer: The combined car and truck move to the right at 1.71 m/s. (Since the result is positive, it matches our initial direction assignment).
  • Key Point: In a perfectly inelastic collision, objects stick together, so they share a single final velocity.

• Example 2: Elastic Collision (Objects Bounce Off)

  • Problem: A 1.0 kg ball moving at 4.0 m/s collides head-on with a stationary 2.0 kg ball. After the collision, the 1.0 kg ball rebounds at 1.0 m/s. What is the velocity of the 2.0 kg ball after the collision?
  • Solution:
    • System: Ball 1 + Ball 2.
    • Initial Momentum (Ball 1): p1_initial = 1.0 kg * 4.0 m/s = 4.0 kg·m/s (right, +).
    • Initial Momentum (Ball 2): p2_initial = 2.0 kg * 0 m/s = 0 kg·m/s.
    • Total Initial Momentum: Σp_initial = 4.0 kg·m/s.
    • Final Momentum (Ball 1): Rebounds at 1.0 m/s, but now moving left (opposite to initial direction). So, v1_final = -1.0 m/s. p1_final = 1.0 kg * (-1.0 m/s) = -1.0 kg·m/s.
    • Final Momentum (Ball 2): Let v2_final be its velocity (direction unknown).
    • Total Final Momentum: Σp_final = p1_final + p2_final = -1.0 kg·m/s + (2.0 kg * v2_final).
    • Conservation: Σp_initial = Σp_final => 4.0 kg·m/s = -1.0 kg·m/s + 2.0 kg * v2_final.
    • Rearrange: 2.0 kg * v2_final = 4.0 kg·m/s + 1.0 kg·m/s = 5.0 kg·m/s.
    • Solve: v2_final = 5.0 kg·m/s / 2.0 kg = 2.5 m/s.
    • Answer: The 2.0 kg ball moves to the right at 2.5 m/s after the collision. (Positive sign indicates the same direction as the initial motion of Ball 1).

• Example 3: Explosion (Initial Momentum Zero)

  • Problem: A stationary 3.0 kg bomb explodes into two pieces. Piece A has a mass of 1.5 kg and moves to the left at 4.0 m/s. What is the velocity of piece B?
  • Solution:
    • System: Bomb fragments (Piece A + Piece B).
    • Initial Momentum: The bomb is stationary, so Σp_initial = 0 kg·m/s.

• Final State: Piece A has a mass of 1.5 kg and a velocity of -4.0 m/s. Let the mass of Piece B be m_b = 3.0 kg - 1.5 kg = 1.5 kg. Let the velocity of Piece B be v_b (direction unknown). * Total Final Momentum: Σp_final = (1.5 kg * -4.0 m/s) + (1.5 kg * v_b). * Conservation: Σp_initial = Σp_final => 0 kg·m/s = (1.5 kg * -4.0 m/s) + (1.5 kg * v_b). * Solve: 1.5 kg * v_b = 6.0 kg·m/s => v_b = 6.0 kg·m/s / 1.5 kg = 4.0 m/s. * Answer: Piece B moves to the right at 4.0 m/s. (Since the result is positive, it matches our assumed direction).

  • Key Point: In an explosion, the total momentum of the system remains zero before and after the event.

Conclusion:

These examples demonstrate the fundamental principle of conservation of linear momentum. Whether dealing with perfectly inelastic collisions, elastic collisions, or explosions, the total momentum of a closed system remains constant, provided no external forces act upon it. Understanding and applying this principle is crucial for analyzing a wide range of physical scenarios, from simple collisions to complex astrophysical events. The key lies in carefully defining the system under consideration, identifying the initial and final states, and applying the conservation equation. By mastering this concept, one can gain a deeper understanding of how motion and momentum are interconnected and predictable within a closed system. Furthermore, these examples highlight the importance of considering direction when dealing with momentum, using positive and negative signs to accurately represent the velocity of objects. This seemingly simple detail is essential for obtaining correct results and interpreting physical phenomena.