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Momentum is a key concept in physics that describes the amount of motion an object has, which depends on its mass and velocity. You might think of it as a measure of inertia in motion. Formally, momentum \( p \) is calculated as \( p = mv \), where \( m \) is the mass and \( v \) is the velocity of the object.

- For the bus in the problem, initially moving with velocity \( v \), its momentum is \( mv \). After some passengers leave, reducing its mass, the momentum changes. If it moves with twice the velocity \( 2v \), but its kinetic energy remains doubly boosted rather than quadruple, the momentum equation helps us understand the reduction in mass affects this balance.

• Before passengers leave, the momentum: \( p_1 = mv \)
• After passengers leave, new momentum: \( p_2 = m'(2v) \)

These relations are crucial in understanding how both the mass and speed affect the momentum and ultimately the de Broglie wavelength.

Kinetic energy is the energy an object possesses due to its motion. It's a key factor in understanding how energy changes as an object moves faster or slower. Calculated as \( KE = \frac{1}{2} mv^2 \), kinetic energy depends on the square of the velocity and directly on the mass.

In the given problem, when the bus's speed doubles, its kinetic energy only doubles in value, not quadruples as one might initially expect because the bus has lost some mass:

• Original kinetic energy: \( KE_1 = \frac{1}{2} mv^2 \)
• New kinetic energy (double): \( KE_2 = \frac{1}{2} m'(2v)^2 = 2 \times KE_1 = mv^2 \)

To achieve this, the mass is halved \( m' = \frac{m}{2} \). These calculations show how changes in mass and velocity influence kinetic energy and help determine changes in the de Broglie wavelength.

Planck's constant \( h \) is a fundamental constant in physics that plays a crucial role in the quantum world. Its value, approximately \( 6.626 \times 10^{-34} \) Js, is what links the de Broglie wavelength equation to momentum: \( \lambda = \frac{h}{mv} \). The constant itself embodies the small, discrete packets of energy that particles in the quantum realm like to work with.

In this exercise, Planck's constant remains unchanged, acting as a scaling factor for connecting momentum and wavelength. When the bus's velocity doubles and mass decreases, it's \( h \) that ensures the product of mass and speed perfectly scales the wavelength accurately.

• Original wavelength: \( \lambda = \frac{h}{mv} \)
• New wavelength: \( \lambda' = \frac{h}{m'(2v)} \)

Planck's constant is a reminder of the consistency needed to bridge between energy, momentum, and wavelength, highlighting the quantum nature even in macroscopic objects like a bus.