The Relation Between Force and Change in Momentum

The relation between force and change in momentum is described by Newton’s Second Law of Motion and can be expressed mathematically as: F=ΔpΔtF = \frac{\Delta p}{\Delta t}

where:

• FF: Force applied on the object (vector quantity)
• Δp=pf−pi\Delta p = p_f – p_i: Change in momentum (pp) over time
• Δt\Delta t: Time interval during which the force acts
• p=mvp = mv: Momentum, where mm is mass and vv is velocity

This equation states that the force is equal to the rate of change of momentum of an object.

Detailed Explanation

1. Momentum (pp): Momentum is the product of an object’s mass and velocity. It is a vector quantity, meaning it has both magnitude and direction.
2. Change in Momentum (Δp\Delta p): When an object’s velocity changes due to an applied force, its momentum changes. The change in momentum can occur in:
   • Magnitude (speeding up or slowing down)
   • Direction (changing the trajectory)
3. Force and Time: The longer the force acts (Δt\Delta t), the greater the change in momentum for the same force. This concept is captured in the impulse-momentum theorem, which states: FΔt=ΔpF \Delta t = \Delta p Here, FΔtF \Delta t is the impulse, a measure of the force applied over a time interval.
4. Special Cases:
   • If the mass is constant, the relationship simplifies to: F=mΔvΔtF = m \frac{\Delta v}{\Delta t} where ΔvΔt\frac{\Delta v}{\Delta t} is the acceleration (aa): F=maF = ma

Example Problem

A ball of mass 2 kg2\,\text{kg} is moving at 5 m/s5\,\text{m/s}. A force is applied, bringing it to rest in 2 s2\,\text{s}. What is the force?

1. Calculate the initial momentum: pi=mvi=(2)(5)=10 kg\cdotpm/sp_i = m v_i = (2)(5) = 10\,\text{kg·m/s}
2. Calculate the final momentum: pf=mvf=(2)(0)=0 kg\cdotpm/sp_f = m v_f = (2)(0) = 0\,\text{kg·m/s}
3. Find the change in momentum: Δp=pf−pi=0−10=−10 kg\cdotpm/s\Delta p = p_f – p_i = 0 – 10 = -10\,\text{kg·m/s}
4. Use the force formula: F=ΔpΔt=−102=−5 NF = \frac{\Delta p}{\Delta t} = \frac{-10}{2} = -5\,\text{N}

The negative sign indicates that the force is in the opposite direction of the ball’s initial motion.