Impulse Knowledge-Based Study

Study notes

1. What is Impulse?

   In physics, impulse describes the overall effect of a force acting on an object over a specific interval of time. When an external force acts on an object, the duration of that interaction is just as critical as the magnitude of the force itself.

   Impulse is mathematically defined as the product of the average net force and the time duration during which the force acts:

   $\vec{I} = \vec{F}_{\text{avg}} \Delta t$

   • $\vec{I}$: Impulse vector
   • $\vec{F}_{\text{avg}}$: Average net force acting on the body
   • $\Delta t$: Time interval during which the force acts

   Properties of Impulse

   • Vector Quantity: Impulse possesses both magnitude and direction. The direction of the impulse vector is always identical to the direction of the applied net force vector.
   • SI Unit: The standard International System of Units (SI) unit for impulse is the newton-second ($\text{N}\cdot\text{s}$), which is dimensionally equivalent to the kilogram-meter per second ($\text{kg}\cdot\text{m/s}$).

   Connecting Impulse to Momentum

   The relationship between impulse and momentum is derived directly from Newton's second law of motion. Expressed in terms of rate of change of momentum, Newton's second law states:

   $\vec{F}_{\text{net}} = \frac{\Delta \vec{p}}{\Delta t}$

   Multiplying both sides of the equation by the time interval $\Delta t$ yields the Impulse-Momentum Theorem:

   $\vec{I} = \Delta \vec{p}$

   Expanding the change in momentum for an object of constant mass $m$:

   $\vec{I} = m\vec{v}_f - m\vec{v}_i$

   • $\vec{v}_i$: Initial velocity vector
   • $\vec{v}_f$: Final velocity vector
   • $\Delta \vec{p}$: Change in linear momentum

   This fundamental theorem states that the impulse delivered to a particle by a net force equals the change in the particle's linear momentum. Because momentum and impulse are vectors, direction must always be accounted for by using consistent sign conventions in one-dimensional problems or vector components in two dimensions.

   Analyzing Variable Forces

   In most real-world interactions, such as a bat hitting a baseball or a collision between two vehicles, the contact force is not constant over time; it increases rapidly from zero to a maximum peak value and then decreases back to zero as contact is broken.

   When force is plotted on the vertical axis against time on the horizontal axis (a Force-Time graph):

   • The area under the curve of a Force vs. Time graph represents the impulse imparted to the object.
   • For a constant force, this area is a simple rectangle: $\text{Area} = F \Delta t$.
   • For linearly varying forces, geometric shapes such as triangles or trapezoids are evaluated to determine impulse.

   The concept of average force is defined such that a constant force of magnitude $F_{\text{avg}}$ acting over the exact same time duration $\Delta t$ delivers the same impulse—and therefore encloses the same area—as the actual time-varying force.

   Controlling Impact Force

   According to the impulse-momentum theorem, bringing a moving object to rest requires a fixed amount of impulse equal to $|\Delta p| = m v_i$. Because:

   $\vec{F}_{\text{avg}} = \frac{\Delta \vec{p}}{\Delta t}$

   The average impact force experienced by an object is inversely proportional to the duration of the impact:

   • Increasing Impact Duration: If $\Delta t$ is made larger, the resulting average force $F_{\text{avg}}$ is significantly reduced. This is the foundational principle behind automobile safety features such as airbags, crumple zones, and padded dashboards, as well as athletic gear like boxing gloves and foam landing mats.
   • Decreasing Impact Duration: If $\Delta t$ is very small, such as when an object strikes a rigid, unyielding surface like concrete, the impact force becomes extremely large, leading to severe structural damage or injury.

   Example Application

   Consider a $0.2\text{ kg}$ ball moving at $20\text{ m/s}$ that rebounds straight back at $15\text{ m/s}$. Taking the initial direction as positive:

   $\Delta p = m(v_f - v_i) = 0.2(-15 - 20) = 0.2(-35) = -7.0\text{ N}\cdot\text{s}$

   The impulse delivered to the ball is $7.0\text{ N}\cdot\text{s}$ in the direction opposite to its initial motion. If the collision lasts $0.02\text{ s}$, the average force exerted on the ball is $F_{\text{avg}} = \frac{-7.0}{0.02} = -350\text{ N}$.