Impacts An impact occurs when two objects are in contact for a very short period of time. 4/15/2017 Dr. Sasho MacKenzie - HK 376

Contact vs. Impacts In general, if two objects are in contact long enough to precisely measure the forces between the objects throughout the contact time, then it is NOT considered an impact. Jumping: foot-ground or Pitching: hand-ball If the contact time is very small and the forces cannot be directly measured, then it is considered an impact. Golf: ball-clubhead or Baseball: bat-ball 4/15/2017 Dr. Sasho MacKenzie - HK 376

Impacts or Collisions Impacts are commonly referred to as collisions in physics. Collisions can be elastic, meaning they conserve energy and momentum, inelastic, meaning they conserve momentum but not energy, or totally inelastic (or plastic), meaning they conserve momentum and the two objects stick together. 4/15/2017 Dr. Sasho MacKenzie - HK 376

Impacts When analyzing a contact as an impact, it is assumed that the collision is an instantaneous event. For example, when a ball bounces off the floor, we would consider the ball to have negative velocity just before impact and then positive velocity the next instant. Impact analysis requires combining three concepts Relative Velocity, Coefficient of Restitution, and Conservation of Momentum. 4/15/2017 Dr. Sasho MacKenzie - HK 376

Relative Velocity When dealing with impact, we need to know what the velocities of the objects are relative to each other. To determine relative velocity, you subtract one velocity from the other. Essentially, you are saying if one object’s velocity was zero, what would the velocity of the other object be? Vbat – vball (example for baseball hitting) The relative velocity of “impact” is simply the absolute value of the above equation. 4/15/2017 Dr. Sasho MacKenzie - HK 376

Relative Velocity: Heading a Soccer Ball -10 m/s +4 m/s Ball relative to Player -10 - (+4) = -14 m/s Player relative to Ball +4 - (-10) = +14 m/s 4/15/2017 Dr. Sasho MacKenzie - HK 376

Coefficient of Restitution (e) It is a measure of the elasticity of two objects that impact. Values can range from 0 (no bounce) to almost 1 (highest bounce). It is calculated using the ratio of the relative velocity of separation to the relative velocity of approach. e = - ( V – v ) ..… 1 ( V – v ) It can also be measured as the ratio of the restitution impulse to the compression impulse. 4/15/2017 Dr. Sasho MacKenzie - HK 376

Conservation of Linear Momentum The forces of impact are internal to the system (M + m). The above law states that if no external forces act on the system, then the momentum of the system is the same before and after impact. MV + mv = MV + mv ….. 2 Combining equations 1 and 2, we can solve for the velocities of the objects post impact. v = M[ V(1+e) – ev] + mv ….. 3 M + m 4/15/2017 Dr. Sasho MacKenzie - HK 376

Special Case v = MV(1+e) M + m In certain cases, when one of the objects is at rest prior to impact (such as golf), the equation can be simplified. The velocity of the golf ball is: v = MV(1+e) M + m Where: v is the velocity of the golf ball after impact, m is the mass of the golf ball, V is the velocity of the clubhead prior to impact, M is the mass of the clubhead, and e is the COR. 4/15/2017 Dr. Sasho MacKenzie - HK 376

Applying the Formula Would a golf ball go farther if you increased clubhead mass, or increased it’s velocity? We can graphically display how each of these variables would affect the velocity of the golf ball after impact. 4/15/2017 Dr. Sasho MacKenzie - HK 376

Increasing Clubhead Mass Max Ball Velocity v = MV(1+e) M + m Ball Velocity Clubhead Mass M is on both the top and bottom of the equation, so when M gets large, increases in M show very small changes in ball velocity 4/15/2017 Dr. Sasho MacKenzie - HK 376

Increasing the Velocity of the Club Ball velocity increases as club velocity increases. Ball Velocity v' = MV(1+e) M + m Velocity of Golf Club V is only on the top of the equation, so the bigger it gets, the bigger ball velocity gets. 4/15/2017 Dr. Sasho MacKenzie - HK 376

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What is the Clubhead Velocity? With what velocity must a 0.25 kg clubhead contact a 0.05 kg golf ball in order to send the ball off the tee with a velocity of 80 m/s? e = .81 v = MV(1+e) M + m V = v (M+m) M (1+e) Therefore, V = 80 (.25+.05) .25 (1 + .81) V = 53 m/s 4/15/2017 Dr. Sasho MacKenzie - HK 376

The Equations of Impact Equations 1 and 2 combine to give … 4/15/2017 Dr. Sasho MacKenzie - HK 376

A golf ball is dropped from a height of 2 m onto the floor A golf ball is dropped from a height of 2 m onto the floor. If it rebounds with a velocity of 3.15 m/s, then what was the coefficient of restitution (e) associated with the impact? Before (e) can be determined, the velocity with which the ball hit the floor must be known, then we can apply the formula e = -v/v 4/15/2017 Dr. Sasho MacKenzie - HK 376

Finding impact velocity of dropped ball If we can figure out how long it takes a ball to fall 2 m, then we can find velocity knowing that v=at, and a, in this case is -9.81 m/s2. Plotting a graph of the ball’s velocity allows us to determine the time that it spends accelerating to the floor. 4/15/2017 Dr. Sasho MacKenzie - HK 376

time of ball impact with floor Since acceleration is a constant -9.81, that means that that the velocity curve will have a constant negative slope of -9.81. velocity t time of ball impact with floor V = at = -9.81t The area under the velocity curve yields displacement. Since the area of a triangle is ½ height x base, we get … d = ½ (-9.81t)(t) = -4.905t2 . Since d=2, we can find t 2 = -4.905t2, therefore, t = 0.64 s. Since Vf = Vi + at, Vf = 0 + (-9.81)(0.65) = -6.3 m/s 4/15/2017 Dr. Sasho MacKenzie - HK 376

Knowing that the golf ball hit the floor with a velocity of -6 Knowing that the golf ball hit the floor with a velocity of -6.3 m/s and rebounded with a velocity of 3.15 m/s, the coefficient of restitution can be calculated e = -v/v = -3.15/-6.3 = 0.5 4/15/2017 Dr. Sasho MacKenzie - HK 376

Baseball Example In which case will the baseball leave the bat with a greater velocity and thus travel farther? The pitcher throws the ball at -40 m/s and the batter swings at 20 m/s. The pitcher throws the ball at -20 m/s and the batter swings at 40 m/s. We could use the impact equations to plug in numbers and find the answer or we could think our way through it. 4/15/2017 Dr. Sasho MacKenzie - HK 376

Relative Velocity of Impact In both examples, the relative velocity of impact is going to be the same. V – v = 20 – (-40) = 60 m/s V – v = 40 – (-20) = 60 m/s The objects do not know what their velocities are, so the characteristics of impact will be the same. The same forces, for the same period of time in both A and B. 4/15/2017 Dr. Sasho MacKenzie - HK 376

Impulse during Impact Impulse = Ft If the forces are the same and the length of time the forces act are the same, then the impulses must be the same for both cases. Impulse = change in momentum = mv Therefore, the change in velocity of the ball (v) equals the Impulse divided by the ball’s mass Ft = mv v = Ft / m 4/15/2017 Dr. Sasho MacKenzie - HK 376

Ball’s Change in Velocity (v) If the impulse is the same for both A and B, then the v is the same for both A and B. Since final velocity is equal to initial velocity plus change in velocity, we can come to a conclusion. vf = vi + v, where v = Ft / m vf = vi + Ft / m 4/15/2017 Dr. Sasho MacKenzie - HK 376

Conclusion vf = vi + Ft / m vf = -40 + Impulse/mass Same for both A and B The ball’s initial velocity is different in A and B. vf = -40 + Impulse/mass vf = -20 + Impulse/mass In case B, the ball will leave the bat with a greater positive velocity, and thus travel further. 4/15/2017 Dr. Sasho MacKenzie - HK 376