Define impulse as force × time for which force acts; recall and use the equation impulse = F Δt = Δ(m v)

Momentum – Impulse

Objective

Define impulse as force × time for which force acts; recall and use the equation impulse = \$F\,\Delta t = \Delta(mv)\$.

Key Concepts

• Impulse (I) is the integral of force over the time interval during which the force acts: \$I = \displaystyle\int{t1}^{t_2} F\,dt\$.
• Impulse is numerically equal to the change in linear momentum: \$I = \Delta p = m\,\Delta v\$.
• Units: \$\text{N\,s}\$ (newton‑seconds) or \$\text{kg\,m\,s}^{-1}\$.
• Impulse is a vector; its direction is the same as that of the net force.
• In collisions, the impulse delivered to an object is equal and opposite to that delivered to the other object (Newton’s third law).

Impulse–Momentum Theorem

The impulse–momentum theorem states that the impulse applied to a body equals the change in its momentum:

\$I = \Delta p = m\,\Delta v = m(vf - vi)\$

When the mass is constant, impulse depends only on the change in velocity.

Example Problem

1. A 0.5 kg ball is moving at 4 m s⁻¹ and collides elastically with a wall, reversing its velocity. Calculate the impulse delivered by the wall.
2. Initial momentum: \$p_i = 0.5 \times 4 = 2.0\ \text{kg\,m\,s}^{-1}\$.
3. Final momentum: \$p_f = 0.5 \times (-4) = -2.0\ \text{kg\,m\,s}^{-1}\$.
4. Change in momentum: \$\Delta p = pf - pi = -2.0 - 2.0 = -4.0\ \text{kg\,m\,s}^{-1}\$.
5. Impulse: \$I = \Delta p = -4.0\ \text{N\,s}\$ (negative sign indicates direction opposite to initial motion).

Summary Table

\$\Delta p\$

Suggested Diagram