Necessity of Defining Impulse

When a force acts on an object for a very short time, it becomes difficult to measure both the force and the time separately. However, the change in momentum caused by this force can be easily measured. Impulse is a special physical quantity we use to describe such situations. It represents the combined effect of force and the time for which it acts. Instead of dealing with immeasurable force and time values, we simply measure the change in momentum. This concept helps us understand collisions, impacts, and other sudden events in physics.

Impulse (J) = F × Δt = Δp = m(v - u)

Where:

• J = Impulse
• F = Force applied
• Δt = Time interval for which the force acts
• Δp = Change in momentum
• m = Mass of the object
• v = Final velocity
• u = Initial velocity

In everyday situations, forces often act for extremely small time intervals. For example:

• When hitting a ball with a bat
• When kicking a football
• When hammering a nail
• When bouncing a ball on a hard surface

In these cases, it is almost impossible to measure the force and time independently because they are too small. However, the effect of these forces (change in momentum) is easily measurable and recordable.

Graphical representation of the impulse of a force

The area under the Force-time graph represents impulse. 

• Before impact: Force is zero
• During impact: Force rises to maximum
• After impact: Force returns to zero
• Shaded area = Impulse = F × Δt

For a constant force: The graph is a rectangle, and impulse = F × Δt

For a varying force: The area under the curve still gives the impulse

• Helps explain sudden collisions and impacts in simple terms
• Allows us to calculate momentum change without knowing the exact force values
• Used in safety design (airbags, cushioning, etc.) to reduce force by increasing time
• Important in sports (catching balls, hitting, kicking)
• Helps understand why wicketkeepers move their hands backward while catching
• Used in engineering and vehicle crash analysis
• Makes calculations easier for extremely brief interactions
• Explains Newton's third law during collisions
• Essential for understanding elastic and inelastic collisions

Given Information:

• Mass of Oxygen molecule (m₁) = 5.35 × 10⁻²⁶ kg
• Mass of Nitrogen molecule (m₂) = 4.65 × 10⁻²⁶ kg
• Velocity of O₂ molecule (u₁) = 400 m/s
• Velocity of N₂ molecule (u₂) = -500 m/s (opposite direction)
• Collision type: Elastic collision
• Collision time (Δt) = 1 ms = 10⁻³ s

Find Final Velocities

For an elastic collision, use the formulas:

\[\mathbf{v}_1=\left(\frac{m_1-m_2}{m_1+m_2}\right)u_1+\left(\frac{2m_2}{m_1+m_2}\right)u_2\] and 

\[\mathrm{v}_2=\left(\frac{m_2-m_1}{m_1+m_2}\right)u_2+\left(\frac{2m_1}{m_1+m_2}\right)u_1\]

Impulse of each molecule:
J₀ = m₀(v₁ - u₁) = 5.35 × 10⁻²⁶ × (-437 - 400) = -4.478 × 10⁻²³ N·s
J_N = m_N(v₂ - u₂) = 4.65 × 10⁻²⁶ × (463 - (-500)) = +4.478 × 10⁻²³ N·s

Net impulse = 0 (Equal and opposite impulses, confirming Newton's third law)
Average force = \[\frac {Impulse}{Time}\] = \[\frac {J}{Δt}\]
F₀ = -4.478 × 10⁻²⁰ N
F_N = +4.478 × 10⁻²⁰ N 

The forces are equal in magnitude but opposite in direction, confirming Newton's third law of motion (action-reaction pair).