A 42 -g firecracker is at rest at the origin when it explodes into three pieces. The first, with mass 12 g, moves along the \(x\) -axis at \(35 \mathrm{m} / \mathrm{s} .\) The second, with mass \(21 \mathrm{g}\), moves along the \(y\) -axis at \(29 \mathrm{m} / \mathrm{s} .\) Find the velocity of the third piece.

Momentum is a fundamental concept in physics, defined as the product of an object's mass and its velocity. When it comes to explosions, such as a firecracker bursting into several pieces, momentum plays a critical role in determining the movement of each fragment.

Consider the firecracker problem: a stationary 42-gram firecracker explodes into three parts, with known masses and velocities for two pieces. The law of conservation of momentum dictates that the total momentum of the system before the explosion must equal the total momentum after the explosion.

Significance of Momentum in Explosions

During an explosion, the sudden release of energy causes the object to split apart, sending pieces flying. Although the firecracker's pieces move in different directions and with different speeds, their momenta combine to equal the initial momentum, which in this case is zero, as the firecracker was initially at rest.

This concept is pivotal because it allows us to determine unknown quantities, such as the velocity of the third piece of the firecracker, by ensuring the momentum in both the x and y directions remains conserved.

Solving physics problems requires a systematic approach, beginning with understanding the concepts involved, identifying knowns and unknowns, and then applying appropriate physical laws to find the solution.

In the case of the exploding firecracker, a step-by-step method guides us through the problem. The initial state of the system is known: the firecracker is at rest, so the initial momentum is zero. After identifying the final momenta for the first two pieces, the principles of physics are employed to infer the third piece's momentum.

Steps for Effective Problem Solving

• Carefully read the problem to understand the physical situation.
• List the known and unknown variables, convert units if necessary.
• Choose the relevant physical principles. Here, it's conservation of momentum.
• Apply mathematical equations to relate the knowns and unknowns.
• Carry out the calculations, being mindful of the direction when dealing with vector quantities.

Ultimately, physics problem-solving is much more than just plugging numbers into formulas; it's about comprehending the underlying principles to apply them correctly to varied situations.

The linear momentum conservation principle states that if no external force acts on a closed system, the total momentum of that system remains constant. This is not only a theoretical concept but also applies to a myriad of practical scenarios, including explosions, collisions, and movements in space.

In our firecracker example, the total momentum before and after the explosion must be equal since no external forces are acting on the pieces. Since we know the momenta of the first and second pieces, we can calculate the momentum of the third piece by balancing the momentum equation for the system.

Applying Momentum Conservation

Applying this principle involves making sure that the total momentum in all directions is conserved. For instance, if the first piece has a momentum in the x-direction and the second in the y-direction, the third piece's momentum must counterbalance these to uphold the total momentum of zero. This final stage of the problem leads to the discovery of the third piece's velocity, showcasing how the conservation of momentum is essential in predicting the outcome of dynamic events.