Force and Momentum

What is Momentum?

• The effect of two colliding bodies on each other depends on their initial velocities and masses.
• Isaac Newton realized that a force was necessary to change the velocity of an object,
  • He defined the momentum of a moving object as mass multiplied by velocity. He showed how the momentum of an object changes when a force is applied to it,
  • and his laws continue to provide essential math rules for predicting object motion, except in quantum physics and strong gravitational fields.
• Newton’s laws are over 300 years old and are still used today, such as in the careful planning of rocket launches using his laws of motion and law of gravitation.
• However, Newton’s laws do not predict the existence of black holes, which were a confirmed prediction of Einstein’s theory of General Relativity.
  • Einstein’s theories of relativity simply extend Newton’s laws where gravity is weak,
  • and the speed of objects is much less than the speed of light.
• Momentum is a vector quantity defined as mass multiplied by velocity,
  • with units of kg m/s and symbol P,
  • For an object of mass m moving at velocity v, its momentum is P=mv,
  • and its direction is the same as the direction of the object’s velocity.

How do Newton’s Laws of Motion relate to Momentum?

Newtons’s first law of motion states that an object will remain at rest or move at a constant velocity unless acted upon by a force.

• Essentially, this law implies that a force is required to alter the momentum of an object.
• When the momentum of an object is steady, it indicates that no resultant force is acting on it.
• If the mass of an object is constant and its momentum is steady, then its velocity must also be steady.
• However, if a moving object with constant momentum gains or loses mass, its velocity would change to keep the momentum constant.
• For instance, in a cycling race, a cyclist who picks up a water bottle would gain mass and lose velocity to maintain the same momentum.

Newton’s second law of motion states that the rate of change of momentum of an object is directly proportional to the resultant force acting upon it, or in other words, the resultant force is equal to the change in momentum per second.

The mathematical representation of Newton’s second law is F=ma, which is derived as follows for an object with a constant mass m under a constant force:

• The acceleration of the object results in a change in its velocity from u to v over a time t, without altering its direction.
• The initial momentum of the object is mu.
• The final momentum of the object is mv.
• The change in momentum of the object is calculated as mv – mu.
• According to Newton’s second law, the force exerted on the object is directly proportional to the change in momentum per unit time.
• Therefore, the force (F) can be expressed as F = (mv – mu) / t = m(v – u) / t = ma, where a = (v – u) / t.
• Newton’s second law of motion can be expressed as a proportional relationship between force, acceleration, and mass, such that F = kma, where k is a constant of proportionality.
• The value of k can be determined by defining the unit of force as the Newton, which is the amount of force required to accelerate an object of mass 1kg by 1m/s². Therefore, k = 1.
• When k = 1, the equation F = ma can be derived from Newton’s second law, assuming that the mass of the object remains constant.
• Generally, the change in momentum of an object can be represented as Δ(mv).
• As a result of the relationship between force and change in momentum, we can express the force F on an object as F = Δ(mv)/ Δt.

1. If the mass of the object is constant, then we can substitute Δ(mv) for mΔv to get F = Δ(mv)/ Δt, where a = Δv/Δt.
2. When mass is transferred continuously at a constant velocity, causing the mass to change at a constant rate, we can express Δ(mv) as vΔm. This means that Newton’s second law can be used in any situation where an object continuously gains or loses mass.

The impulse of a force is the product of the force and the time for which it acts. We can express it as Impulse = FΔt = Δ(mv).