Understanding Momentum
From Newton’s first law, we know that a moving object keeps moving in a straight line with constant velocity unless acted upon by an external force. The magnitude and direction of the velocity remain unchanged.
Now, suppose two objects of different masses are moving with the same velocity. We see that to bring a heavy object to rest, more force is required than a light object. Therefore, the force required to stop a moving object depends on the mass of that object. Again, it is seen from experience that the force required to stop the object also depends on the velocity of the object. That is if two objects of the same mass move with different velocities, then to stop the faster object, more force is required.
That is, the force required to stop a moving object depends on two quantities. They are: (i) the mass of the object and (ii) the velocity of the object. In this context, Newton introduced a new quantity. that is momentum. Momentum is equal to the product of the mass of the moving object and its velocity.
Another example is two trucks, one is fully loaded and the other is empty striking a wall with the same velocity. Which truck will destroy the wall the most? Of course the loaded truck. Now two unloaded trucks, one has faster velocity, strike the wall. In this case, which truck will destroy the wall the most? Here the answer is that truck with faster velocity.
So from the above example, we can conclude that not only the velocity alone, or the mass alone will impact the wall. The combined effect of mass and velocity is responsible for that. This combination of mass and velocity gives a new type of physical quantity, called momentum. Momentum is a fundamental concept in physics.
Definition of Momentum
The physical quantity that arises in a moving object by the combination of mass and velocity is called linear momentum.
It is also defined as the product of an object’s mass and velocity.
Formula
Momentum (p) = Mass (m) × Velocity (v)
Or as a vector form –
$$\text{Momentum}\ (\vec{p}) = \text{Mass} (m) \times \text{Velocity} (\vec{v})$$
$$\vec{p} = m \cdot \vec{v}$$
Nature
Momentum is a vector quantity as it has both magnitude and direction. It has the same direction as the direction of velocity.
Unit
As we know, Momentum = Mass × Velocity
So unit of momentum = unit of mass × unit of velocity
- SI or MKS unit of momentum = Kg.m/s or Kg.m.s−1
- CGS unit of momentum = g.cm/s or g.cm.s−1
Dimension
We know, Momentum = Mass × Velocity
So Dimension of momentum = Dimension of mass × Dimension of velocity
Dimension of momentum = [M] × [LT−1]
Dimension of momentum = [MLT−1]
Newton’s Second Law and Momentum
Newton’s Second Law of Motion states that the force acting on an object is equal to the rate of change of its momentum. Mathematically:
$$\vec{F} = \frac{d\vec{p}}{dt}$$
- F = force (N)
- p = momentum (kg·m/s)
- t = time (s)
From the definition of momentum (p=mv), differentiate with respect to time:
$$ \vec{F} = \frac{d\vec{p}}{dt} = \frac{d(m\vec{v})}{dt}$$
If mass m is constant:
$$\frac{d\vec{p}}{dt} = m \frac{d\vec{v}}{dt} = m\vec{a}$$
$$\vec{F} = m\vec{a}$$
This is the Newton’s Second Law.
Impulse-Momentum Theorem
Impulse is the change in momentum of an object when a force is applied for a short time. It is given by:
Impulse J = F⋅t = Δp
where:
- F = force applied
- t = time duration
- Δp = change in momentum
From Newton’s Second Law:
$$\vec{F} = \frac{d\vec{p}}{dt}$$
integrate over time
$$\int \vec{F} \, dt = \int d\vec{p}$$
$$\vec{J} = \Delta \vec{p}$$
This shows that the impulse applied to an object equals its change in momentum.
Law of Conservation of Momentum
The law of conservation of momentum states that:
“The total momentum of an isolated system remains constant if no external force acts on it.”
$$\vec{p}_{\text{initial}} = \vec{p}_{\text{final}}$$
Elastic Collision:
In an elastic collision, both momentum and kinetic energy are conserved. For two objects of masses m1 and m2:
$$m_1 \vec{v}_1 + m_2 \vec{v}_2 = m_1 \vec{v}_1′ + m_2 \vec{v}_2’$$
- v1,v2 = initial velocities
- v1′,v2′ = final velocities
Inelastic Collision:
In an inelastic collision, momentum is conserved, but kinetic energy is not. For a perfectly inelastic collision (objects stick together):
$$m_1 \vec{v}_1 + m_2 \vec{v}_2 = (m_1 + m_2) \vec{v}’$$
- v′ = final velocity of the combined mass
Relation Between Kinetic Energy and Momentum
Kinetic energy (K) and momentum (p) are related through the following equation:
K = p2/(2m)
From the definition of kinetic energy:
$$K = \frac{1}{2} m v^2$$
momentum p=mv, or v = p/m. Putting this
$$K = \frac{1}{2} m \left( \frac{p}{m} \right)^2 = \frac{p^2}{2m}$$