Impact

Impact is when 2 object interact and cause a large force (impulse) between them

There are 2 types of impacts:

1. Central Impact: when the motion of the center of mass is along the same line that passes through the center of mass for both objects, also known as the line of impact.
   
2. Oblique Impact: the motion of one, or more, object makes an angle with the line of impact.

During a collision the objects will deform and they will exert an equal and opposite deformation impulse on one another. When maximum deformation occurs both objects will travel with the same velocity. After maximum deformation occurs a period of restitution occurs. During the restitution period the objects will either return to the original shape or stay permanently deformed. This restitution period makes the objects "push" apart. The deformation impulse will be greater than the restitution impulse.

For a collision the conservation of linear momentum ( ∑mv1 = ∑mv2 ) can be used since the internal impulses (deformation and restitution) cancel out the momentum.

A second equation that can be used is to apply the principle of impulse and momentum for each object.

Coefficient of Restitution(e): the ratio of the restitution impulse to the deformation impulse. Also the ratio of the relative velocity when the objects separate after the collision to the relative velocity of the objects before impact.

e = (VB2 - VA2) / (VA1 - VB1)

Where V is the velocity, A is one object, B is another object, 2 is after impact, and 1 is before impact.

The coefficient of restitution has a value between zero and one.

Elastic Impact/Perfectly Elastic (e = 1): This can't be achieved in real life because the restitution impulse will be equal to, but opposite of the deformation impulse. No energy loss.

Plastic Impact/Inelastic (e = 0): This occurs when both objects stick together after impact, and both will have the same velocity. Maximum energy loss.

The energy loss from a collision can be found by finding the difference in the kinetic energy. Change in energy = ∑T2 - ∑T1
Where T = .5 m v^2 and T2 is the final kinetic energy and T1 is the initial kinetic energy.

Energy loss from a collision can be due to sound or thermal energy.


Oblique Impact: after impact there will be 4 values that define the object (2 angles the two objects make and their final velocities). The line of impact is drawn through the center of mass of both objects and is perpendicular to the line of contact. Typically a coordinate system will be set up such that the line of impact and the line of contact will be the x and y axis.
Momentum will be conserved along the line of contact(y-axis) because no impulses act on the objects in that direction, the velocities will not change along that direction.

Impulse & Momentum - Dynamics

The Principle of Linear Impulse & Momentum

The equation of motion (F = ma) will be integrated to give the principle of linear impulse and momentum. This is a vector equation that relates change in the velocities magnitude and direction, also relates force and time.

Linear Momentum: L = m v where m is mass and v is velocity. Units are mass-velocity, Kg*m/s or slug*ft/s

Linear Impulse: the effect of a force is measured over a period of time. I = ∫F dt The units are force-time, N-s or lb-s


∑mv1 + ∑∫F dt = ∑mv2 This is the Principle of linear impulse and momentum

This equation states that the initial momentum plus the impulses of all external forces is equal to the final momentum. This equation can also be written in to the x, y, and z components for the velocity and force.


If the sum of the external impulses that act on the system are equal to zero then the conservation of linear momentum can be written...

∑mv1 = ∑mv2

This equation is helpful when dealing with objects that collide/interact.

Impulsive forces: these forces are present when there is an explosion, or the striking of one object on another.

Non-impulsive forces: a force that is very small compared to the impulsive forces such as weight or a spring that has a small deformity.

Example: A soccer ball that is kicked. The force of the foot kicking the ball is an impulsive force because the momentum of the ball is changed drastically. The balls weight is a non-impulsive force and has a negligible effect on the change in momentum.



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