Impacts and Their Graphs
A closer look at Impulse
Horizontal Impulse in Running
+ Net Positive Accel. During Contact
0 Positive impulse > negative impulse
No Net Acceleration During Contact
0 Positive impulse = negative impulse
Net Negative Accel. During Contact
0 _ Positive impulse < negative impulse
Impulse is the area under the
Time (s) force-time curve.
F = ma
Force (N) Time (s)
F = ma
-force - acceleration
At each instant in time during a contact, a force acts to produce an
acceleration. The Impulse is the net effect of all those
instantaneous forces. In other words, it is the average force
multiplied by the total time over which the forces have acted.
Impulse = Average Force x Time force was applied or Fdt
Impulse produces Mass x Change in Velocity or mv
• During a single running contact, your body
undergoes both positive and negative forces
that produce positive and negative
• A force acting for a period of time produces
• If the positive and negative impulses cancel
each other out (equal areas), then the net
impulse is zero and the runner is moving at
a constant speed.
Equal and Opposite
When a ball hits a bat, or a foot lands on the ground, there
are equal and opposite impulses applied to both objects. The
force-time curve of one is the inverted force time curve of
Peak forces are equal and
occur at the same time
Vertical Impulse: Countermovement jump
Fy = Fy – mg = may
Combining Fy and -mg gives the graph of the sum of the
forces in the vertical direction
Resultant Forces (Fy)
a: Bottom of Squat c d
c: Peak Jump Height
e: Back to Standing
Acceleration of Center of Mass
Where is the greatest negative velocity?
Where is the greatest positive velocity?
At what point is the jumper’s velocity zero?
Vf = Vi + aydt
• The equation states that final velocity is
equal to initial velocity plus the change in
• Where the change in velocity is the net area
under the acceleration curve.
Relative Velocity of Impacting
• 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?
Relative Velocity: Heading a Soccer Ball
Ball relative to Player
-10 - (+4) = -14 m/s
Player relative to Ball
+4 - (-10) = +14 m/s
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
It can also be measured as the ratio of the restitution
impulse to the compression impulse.
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
• 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)
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.
Applying the Formula
• Would a golf ball go farther if you
increased clubhead mass, or increased it’s
• We can graphically display how each of
these variables would affect the velocity of
the golf ball after impact.
Increasing Clubhead Mass
Max Ball Velocity
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
Increasing the Velocity of the Club
Ball velocity increases as
club velocity increases.
Velocity of Golf Club
V is only on the top of the equation, so the bigger it gets, the
bigger ball velocity gets.
Clubhead Mass Clubhead Velocity Ball Mass e Ball Velocity
0.5 40 0.05 0.83 66.5
1 40 0.05 0.83 69.7
2 40 0.05 0.83 71.4
10 40 0.05 0.83 72.8
1000 40 0.05 0.83 73.2
0.5 40 0.05 0.83 66.5
0.5 45 0.05 0.83 74.9
0.5 50 0.05 0.83 83.2
0.5 70 0.05 0.83 116.5
0.5 1000 0.05 0.83 1663.6
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) Therefore,