A particle of mass m moving with a velocity u makes an elastic one-dimensional collision with a stationary particle of mass m establishing a contact with it for extremely small time T. Their force of contact increases from zero to F 0 linearly in time $\frac{T}{4}$ , remains constant for a further time $\frac{T}{2}$ and decreases linearly from F 0 to zero in further time $\frac{T}{4}$ as shown. The magnitude possessed by F 0 is
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Change in momentum = Impulse
= Area under force-time graph
∴ ∴ $m \overline{v}$ Area of trapezium
⇒ ⇒ $mv = \frac{1}{2} \left( T + \frac{T'}{2} \right) F_0$
⇒ ⇒ $m v = \frac{3 T}{4} F_0$ ⇒ ⇒ $F_0 = \frac{4 m u}{3 T}$
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