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Impacts

An impact is defined as a collision that occurs within a very short period of time.  This will result in a relatively large (impulsive) force between the two bodies at the point of collision.  An example of an impact would be when you are hammering a nail, or when a baseball player hits a baseball when he is up to bat.

There are two types of impacts.  The first type of impact is called a central impact.  A central impact occurs when when the center of mass of the two particles is in line with each other during the impact.  On the other hand if the line of impact of the two particles are at an angle to each other this would be called an oblique impact.

Mechanics of an Impact

As seen in the image below, before an impact the particle will moving towards each other at a certain velocity.  This velocity could be the same or it could be different.  There could also be cases where the particles are moving in the same direction, but the particle behind the particle in front would be moving faster which eventually cause an impact.  Once the impact occurs a certain (impulsive) force will occur base on the total time of the impact.  As impact occurs the particles can be expected to deform.  The deformation will cause an opposite but equal deformation impulse ∫P dt.  Once maximum deformation occurs the particle will at that instant be moving at the same relative velocity until they start to move away from each other.  Assuming there is no permanent deformation to either particle, as the particles move away from each other they will return to their original shape.  This is caused by the restitution impulse ∫dt.  After the impact has concluded the velocity and direction of the particles will change, as represented by the equation below.

(Eq 1)  $e=\frac{∫Rdt}{∫Pdt}=\frac{(v_B)_2-(v_A)_2}{(v_A)_1-(v_B)_1}$

(vA)1 = initial velocity of particle A

(vA)2 = final velocity of particle A

(vB)1 = initial velocity of particle B

(vB)2 = final velocity of particle B

e = coefficient of restitution

Coefficient of Restitution

Notice in the equation above there is a variable e called the coefficient of restitution.   The coefficient of restitution is the ratio of the relative velocity of the two particle after impact and the relative velocity of the two particle before impact.  If the collision between the two particle is perfectly elastic than e would equal 1.  On the other hand if the impact was completely plastic meaning the impact causes the particle to become completely stuck together than e would equal 0.  Both these cases are theoretical impacts.  Instead the coefficient of restitution is a value between 0 and 1.

Example

Find the velocity of the two particle after impact occurs.

 

Solution

Step 1: Use equation 1.

$0.75=\frac{(v_B)_2-(v_A)_2}{20-5}$

$11.25 = (v_B)_2-(v_A)_2$

Step 2: The conservation of momentum equation will need to be used to solve for the two unknown variables.

$m_B(v_B)_1+m_A(v_A)_1=m_B(v_B)_2+m_A(v_A)_2$

$3~kg(5~m/s)10~kg(20~m/s)=3~kg(v_B)_2+10~kg(v_A)_2$

$215 = 3~kg(v_B)_2+10~kg(v_A)_2$

Step 3:  Solve for $v_A$ and $v_B$.

$v_A=13.9~m/s$   $v_B=25.2~m/s$

Oblique Impact

So far I have only talked about central impacts.  However, as mentioned above, there can also be cases where there are oblique impacts.  For central impacts we can intuitively determine the direction of particle after impact, and we only have to determine the change in velocity of the particles due to the impact.  This will result in two unknowns $(v_A)_2$ and $(v_B)_2$.  While on the other hand an oblique impact will have four unknowns due to the impact occurring at an angle.  If the initial velocity and angle of impact are known, than the four unknowns would be as follows $(v_A)_2$, $(v_B)_2$, $θ_2$, and Φ_2.  To find these unknowns vector algebra would need to be used.  This can be seen in the image below.

 

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