Chapter 7: Impulse and Momentum

7.1 The Impulse-Momentum Theorem

Impulse: change in momentum

• The impulse of a force is the product of the average force and the time interval during which the forces act

• Equation: \overrightarrow{J}=\overline{\overrightarrow{F}}\Delta t

• SI units: N\cdot s

• Vector quantity

• Same direction as the average force

Definition of Linear Momentum

• The linear momentum of an object is the product of the object’s mass times its velocity.

• “Mass in motion”

• Equation: \overrightarrow{p}=m\overrightarrow{v}

• SI units: \dfrac{kg\cdot m}{s}

• Vector quantity

• Has the same direction as the velocity

Impulse-Momentum Theorem

• When a net force acts on an object, the impulse of this force is equal to the change in the momentum of the object.

• If you apply force on an object, it will gain some momentum.

• An object that loses momentum must be transmitting force.

• If an object is undergoing momentum, it is experiencing an impulse.

• No impulse acting on an object means the momentum is conserved.

• Equation: \left( \sum \overline{\overrightarrow{F}}\right) \Delta t=m\overrightarrow{v}_{f}-m\overrightarrow{v}_{0}

  • Impulse x elapsed time = final momentum + initial momentum

7.2: The Principle of Conservation of Linear Momentum

• The total linear momentum of an isolated system remains constant (is conserved) if no external forces act on it.

• An isolated system is one for which the vector sum of the average external forces acting on the system is zero.

• The impulse–momentum theorem states that the impulse produced by a net force is equal to the change in the object's momentum, whereas the work–energy theorem states that the work done by a net force is equal to the change in the object's kinetic energy.

• Cancellation of the internal forces occurs no matter how many parts there are to the system and allows us to ignore the internal forces.

• Isolated system: a system where the sum of external forces is zero.

• Whether the system is isolated depends on whether the vector sum of the external forces is zero.

• Equation: \overrightarrow{p}_{f}=\overrightarrow{p}_{0}

Internal and External Forces

• Internal forces: Forces that objects within the system exert on each other (cancel due to Newton’s Third Law).

• External forces: Forces exerted on objects by agents external to the system. If sum to 0, then momentum is conserved.

Applying the Principle of Conservation of Linear Momentum

1. Decide which objects are included in the system.

2. Relative to the system, identify the internal and external forces.

3. Verify that the system is isolated (no net external forces acting on it, the forces must sum to zero)

4. Set the final momentum of the system equal to its initial momentum. Remember that momentum is a vector.

5. Check the signs.

7.3: Collisions in One Dimension

Total momentum of isolated systems is conserved in collisions

• In any collision linear momentum is conserved.

• Equation: m_{1}v_{f1}+m_{2}v_{f2}=m_{1}v_{01}+m_{2}v_{02}

• Three types of collisions: elastic, inelastic, and completely inelastic

• Elastic collision example: billiard balls colliding

• Inelastic collision example: a baseball bat hitting a baseball

• Completely inelastic collision example: cars sticking together after impact

• Final velocity after collision equation:

  • v_{f}=\dfrac{m_{1}v_{01}+m_{2}v_{02}}{m_{1}+m_{2}}

7.4: Collisions in Two Dimensions

• Collisions within isolated systems conserve perpendicular components of momentum.

• x component: m_{1}v_{ f1x}+m_{2}v_{f2x}=m_{1}v_{01x}+m_{2}v_{02x}

• y component: m_{1}v_{ f1y}+m_{2}v_{f2y}=m_{1}v_{01y}+m_{2}v_{02y}

7.5: Center of Mass

• Center of mass: a point that represents the average location for the total mass of a system.

  • Equation: x_{cm}=\dfrac{m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}

  • SI units: m

• Total linear momentum of isolated system does not change.

• The velocity of the center of mass does not change.

• If you have an isolated system with a moving center of mass, the velocity of the center of mass does not change.

• Velocity of center of mass

  • Equation: v_{cm}=\dfrac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}

  • SI units: m/s

  • Vector quantity