Universal power-law decay of the impulse energy in granular protectors

According to Newton’s second law of motion, the rate of change of the momentum of an object is the net external force acting on it. The total change in momentum between two timepoints thus depends on both the external force acting on it and the time over which it acts. Describing this mathematically, the total change of an object’s motion is proportional to the force vector and the time over which it is applied. This product is called impulse.
Additionally, it can be shown that the total...

The total change in the motion of an object is proportional to the total force vector acting on it and the time over which it acts. This product is called impulse, a vector quantity with the same direction as the total force acting on the object.
By writing Newton's second law of motion in terms of the momentum of an object and the external force acting on it, and simultaneously using the definition of the impulse vector, it can be shown that the total impulse on an object is equal to its net...

In the context of a system of particles moving relative to an inertial frame of reference, the equation of motion is a crucial tool for understanding the dynamics of the system. This equation, which accounts for external forces acting on each particle, plays a fundamental role in describing the system's behavior.
Notably, internal forces between particles, occurring in equal and opposite collinear pairs, cancel out and are not part of the equation of motion. This exclusion simplifies the...

In the dynamic realm of billiards, a fascinating interplay of forces governs the motion of cue balls and stationary balls. When the cue ball collides with a stationary ball, linear momentum is exchanged. The cue ball imparts a fraction of its linear momentum to the stationary ball, causing the cue ball to decelerate while initiating the motion of the stationary ball.
The impulsive force at play during this interaction is of extremely short duration, rendering its impulse negligible. When...

Impact occurs when two bodies collide, leading to the application of impulsive forces between them. Analyzing impact mechanics involves considering two colliding particles moving along a line known as the line of impact, which passes through their centers and is perpendicular to the contact plane.
When particles with different initial velocities collide, they induce deformation by applying equal and opposite impulses. At the point of maximum deformation, the particles move together with...

When one considers a rigid body undergoing a plane motion, which is essentially a blend of translational and rotational movement, the application of Newton's second law gives the formula for the translational movement of such a body. If this equation is multiplied by a time interval, dt, and then integrated over the limits of integration, it results in an equation that embodies the principle of linear impulse.