Download direction of the reaction force

KINE 3301
Biomechanics of Human Movement
Linear Impulse − Momentum
Applications
Chapter 9
The force shown below is applied to a 3 kg bowling
ball with an initial horizontal velocity of −2 m/s.
Compute the final velocity of the ball.
𝐹 ∆𝑡 = 𝑚𝑉𝑓 − 𝑚𝑉𝑖
(31.65 N) .315𝑠 = (3 𝑘𝑔)𝑉𝑓 −
𝑉𝑓 = +1.33 𝑚/𝑠
What was the impulse?
𝐽 = 𝐹∆𝑡
𝐽 = 31.65 𝑁 (.315𝑠)
𝐽 = 9.97 𝑁 ∙ 𝑠
𝑉𝑓 −
𝑚
(3 𝑘𝑔)(−2 )
𝑠
Integration of the force with
respect to time (area under the
force – time curve) can be used to
obtain the velocity – time curve.
𝑡1
𝐹 𝑑𝑡 = 𝑚𝑉𝑓 − 𝑚𝑉𝑖
𝑡0
The two force curves shown below are applied to a 0.5 kg ball with an
initial horizontal velocity of 0 m/s. Compute the final velocity of the
ball after each force is applied.
Draw an estimated velocity-time curve that each force-time curve
would produce.
Reaction Force Accelerates the CM
The force applied accelerates
the ground in the direction
of the force.
The reaction force
accelerates the performer’s
center of mass in the
direction of the reaction
force.
Relationship between Force & Acceleration
𝐹 = 𝑚𝑎
The shape of an
acceleration curve is
the exactly the same
as the force curve,
only the units are
different.
𝐹
𝑎=
𝑚
Impulse-Momentum
𝐹 ∆𝑡 = 𝑚𝑉𝑓 − 𝑚𝑉𝑖
𝑡1
𝐹 𝑑𝑡 = 𝑚𝑉𝑓 − 𝑚𝑉𝑖
𝑡0
Horizontal Impulse-Momentum
𝐹𝑥 ∆𝑡 = 𝑚𝑉𝑥𝑓 − 𝑚𝑉𝑥𝑖
𝑡1
𝐹𝑥 𝑑𝑡 = 𝑚𝑉𝑥𝑓 − 𝑚𝑉𝑥𝑖
𝑡0
Vertical Impulse-Momentum
𝐹𝑦 ∆𝑡 + 𝑚𝑔 ∆𝑡 = 𝑚𝑉𝑦𝑓 − 𝑚𝑉𝑦𝑖
𝑡1
(𝐹𝑦 + 𝑚𝑔) 𝑑𝑡 = 𝑚𝑉𝑦𝑓 − 𝑚𝑉𝑦𝑖
𝑡0
Use the average force to
compute braking impulse,
propulsion impulse and Vx
at midstance (t = .112 s)
and toe-off (t = .234 s).
Braking and Propulsion
Braking < Propulsion
∆Vx = +.46 m/s
Braking ≈ Propulsion
∆Vx = +.01 m/s
Braking > Propulsion
∆Vx = −.24 m/s
Free Body Diagram for Vertical Impulse - Momentum
𝐹𝑦 ∆𝑡 + 𝑚𝑔 ∆𝑡 = 𝑚𝑉𝑦𝑓 − 𝑚𝑉𝑦𝑖
𝑡1
(𝐹𝑦 + 𝑚𝑔) 𝑑𝑡 = 𝑚𝑉𝑦𝑓 − 𝑚𝑉𝑦𝑖
𝑡0
Use the average force
F Ave = 1007.075 N to
compute the vertical impulse
and Vy at toe-off (t = .234 s).
Walking Forces
Use the average force to compute braking
impulse, propulsion impulse and Vx at
t = 0.04, t = 0.4, and t = 0.7 s.
Use the average force F Ave = 621.88 N to
compute the vertical impulse and Vy at
toe-off (t = 0.76 s).
Vertical Force &
Acceleration for a
Vertical Jump
Use the average force at
each time point to compute
the vertical velocity.
t = 0.2 s, F Ave = 440 N
t = 0.4 s, F Ave = 632 N
t = 0.6 s, F Ave = 904 N
Use the average force at
each time point to compute
the vertical velocity.
t = 0.2 s, F Ave = 440 N
Use the average force at
each time point to compute
the vertical velocity.
t = 0.4 s, F Ave = 632 N
Use the average force at
each time point to compute
the vertical velocity.
t = 0.6 s, F Ave = 904 N
At t = 0.4 sec the jumper
has a vertical velocity (Vyi)
of −0.26 m/s. Use the
average force from t = .4 to
t = .6 to compute the
impulse and the final
vertical velocity at t = 0.6
sec.
∆t = 0.2 s, F Ave = 1449 N