In a scenario where a dart with mass ( m_d …

Physics Questions
Short Answer
Step by Step Solution
Physics Questions

In a scenario where a dart with mass ( m_d ) is launched at a block of mass ( m_b ), which is suspended by a string of length ( L ), the dart strikes the block horizontally at speed ( v ) and becomes embedded in it. The resulting dart-block system swings up to a maximum height ( H ). Given that ( m_b > m_d ): (a) Which object experiences a larger impulse during the collision, the dart or the block? If they experience the same magnitude of impulse, please specify. Provide a brief explanation for your reasoning. (b) If the dart’s speed upon embedding into the block exceeds ( v ), how would the maximum height reached by the center of mass of the dart-block system compare to ( H )? Explain your answer without using equations.

Short Answer

The answer explains impulse as the product of force and time during interactions between objects, emphasizing its equality to ensure momentum conservation. It also discusses energy conservation, highlighting the conversion of kinetic energy to potential energy, and concludes by illustrating the direct relationship between speed and height, where higher speeds result in greater heights.

Step-by-Step Solution

Step 1: Understanding Impulse

Impulse is defined as the product of force and the duration of time that the force acts. When two objects interact, such as a dart hitting a block, the impulse experienced by both is equal. This is because the interaction time between them is constant, leading to the conclusion that:

  • Impulse experienced by the dart = Impulse experienced by the block
  • This equality ensures momentum conservation during the collision.

Step 2: Energy Conservation in the Reaction

In any reaction, especially in collisions, the principle of conservation of energy plays a crucial role. Kinetic energy (energy of motion) converts into potential energy (stored energy) during the interaction. Thus, the equation governing this relationship can be expressed as:

  • Kinetic Energy = Potential Energy
  • Mathematically represented as: 0.5Mv² = gh(md + mb)

Step 3: Relationship Between Speed and Height

The relationship between speed and height shows how kinetic energy can influence potential energy. As the speed of an object increases, so does the height it can reach after the interaction. This means:

  • Higher speeds lead to greater heights.
  • The height is directly proportional to the square of the speed, confirming that increased velocity results in increased height.

Related Concepts

Impulse

The product of force and the duration of time that the force acts, experienced equally by interacting objects during collisions

Conservation Of Energy

A principle stating that energy cannot be created or destroyed, only transformed from one form to another, particularly in reactions and collisions

Relationship Between Speed And Height

The correlation indicating that as the speed of an object increases, the maximum height it can reach also increases, with height being proportional to the square of the speed.

Table Of Contents
  1. In a scenario where a dart with mass ( m_d ) is launched at a block of mass ( m_b ), which is suspended by a string of length ( L ), the dart strikes the block horizontally at speed ( v ) and becomes embedded in it. The resulting dart-block system swings up to a maximum height ( H ). Given that ( m_b > m_d ): (a) Which object experiences a larger impulse during the collision, the dart or the block? If they experience the same magnitude of impulse, please specify. Provide a brief explanation for your reasoning. (b) If the dart's speed upon embedding into the block exceeds ( v ), how would the maximum height reached by the center of mass of the dart-block system compare to ( H )? Explain your answer without using equations.

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