1. Impulse and Momentum
Work & Energy Impulse & Momentum
• Scalar equation. • Vector equation.
• The concept of work relates force to • The concept of impulse relates force to time.
displacement.
• Greater force or greater displacement is • Greater force or greater time of action is
associated with more work done. associated with more impulse applied.
• More work done changes the motion of a • More impulse changes the motion of a system
system to a greater degree. to a greater degree.
• That which is changed is called kinetic • That which is changed is called momentum.
energy. Impulse & Momentum
Work & Energy
Principle of Linear Impulse and Momentum
Consider Newton’s 2nd law.
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2. Components of Impulse
Rectangular coordinate system
• The final momentum of a particle is obtained by adding vectorially its initial momentum
and the impulse of the force F acting during the interval considered.
Conservation of Linear Momentum
When the sum of the external impulses acting on a system of particle is zero, the equation for the
principle of linear impulse and momentum reduces to the following:
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3. Consider two boats initially at rest, which are pulled together
Impulsive forces and Motion
Impulsive force is a force that acts on a particle during a very short time interval and produces a
definite change in momentum. The resulting motion is called an impulsive motion. Baseball
hitting a bat.
Non impulsive forces like weight of the body, the force exerted by spring, or any other force
which is known to be small compared with the impulsive force may be neglected.
• In case of the impulsive motion of several particles, we can write:
• No impulsive external forces acting on the body
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4. Example
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5. Example
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7. Impact
• Impact is a collision of two bodies, in a short interval of time, producing a large force
between them.
• Normal to the tangent line at the contact point is called the line of impact
Direct central impact
• The mass centers of the two
bodies are on the line of impact
• The impact velocities of the
two
bodies are directed along the
line of impact
Oblique central impact
• The impact velocities of the
two
bodies are not directed along
the line of impact
Direct Central Impact
Consider the impact of two particles
The total momentum of the two particles is conserved
Scalar components
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8. Velocities after the Impact & the Coefficient of Restitution (e)
To obtain a second relationship between velocities, consider the motion of particle A and B
during the period of deformation and apply the principle of impulse and momentum.
Particle A
Forces P and R are exerted on particle B and on particle A, in general R is different than P
• The ratio of the magnitude of the impulses corresponding to the period of restitution and to
the period of deformation is called the coefficient of restitution, e is always between 0 and 1.
Substitute for the impulses
Same approach for particle B gives:
Eliminate u
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9. • The following two equations can be used to determine the velocities of the particles after the
impact.
Sign convention
The equations were derived assuming that particle B is located on the right of particle A and both
particles are initially moving to the right. If particle B is moving to the left a negative sign should
be considered. Same sign convention holds for after the impact.
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10. example
A 20,000 kg railroad car moving at a speed of 0.5 m/s to the right collides with a 35,000 kg car
which is at rest. If after the impact the 35,000 kg car is observed to move to the right at a speed
of 0.3 m/s, determine the coefficient of restitution between the two cars.The total momentum of
the two cars is conserved.
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11. Example
A ball is thrown against a frictionless, vertical wall. Just before the ball strikes the wall, its
velocity has a magnitude v and forms an angle of 30o with the horizontal. If the coefficient of
restitution between the ball and wall is 0.9, determine the magnitude and direction of the velocity
of the ball as it rebounds from the wall.
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