3.11 Momentum and Impulse
Suppose that a bicycle rider and a heavy truck are moving with the same speed. Which one can be stopped easily depends on the quantity of motion of the moving body. It is our common observation that the quantity of motion in a moving body depends on its mass and velocity. The greater the mass, the greater the quantity of motion; similarly, the greater the velocity, the greater the quantity of motion. This quantity of motion is called momentum and is denoted by p.
lightbulbThe momentum of a moving body is the product of its mass and velocity.
p = m × v ............ (3.2)
Like velocity, momentum is a vector quantity. The SI unit of momentum is kg m s⁻¹ (or N s).
When a ball is hit by a bat, the force is exerted on the ball for a very short interval of time. In such cases, it is very difficult to calculate the exact magnitude of the force. However, the initial velocity v_i of the ball and the final velocity v_f after collision can be found easily.
During a time interval Δt, the average acceleration a is given by:
a = Δv / Δt = (v_f - v_i) / Δt ............ (3.3)
According to Newton's second law of motion, the value of average force acting during the interval Δt will be:
F = ma = m(Δv / Δt)
or F × Δt = m(Δv) = m(v_f - v_i) ............ (3.4)
Equation (3.4) shows that F and Δt cannot be exactly known, but their product, which is equal to the change of momentum (mv_f - mv_i), can be calculated. For such cases, the product F × Δt is called the impulse of the force.
lightbulbWhen a large force F acts on an object for a short interval of time, the impulse of the force is defined as the total change in momentum of the object.
Dividing both sides of Eq. (3.4) by Δt:
F = m(Δv) / Δt ............ (3.5)
where m(Δv) is the change in momentum Δp. Equation (3.5) gives the value of force in terms of momentum:
F = Δp / Δt ............ (3.6)
Equation (3.6) suggests another definition of Newton's second law of motion in terms of momentum:
lightbulbThe rate of change of momentum of a body is equal to the force acting on it.
The direction of change in momentum is that of the force.