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Wize AP Physics C: Mechanics Textbook > Unit 4: Systems of Particles and Linear Momentum (14-17%)

Linear Momentum and Impulse

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Linear Momentum and Impulse


Linear Momentum

Linear Momentum (or simply Momentum) is defined as the product of mass and velocity of an object.

Momentum is a vector and measured in kg . m/s.

P⃗=mv⃗\boxed{\vec{P}=m\vec{v}}

If a force is applied to an object, the Second law of Newton could be written in terms of change in momentum as follows:

F⃗=ma⃗         →            F⃗=mΔvΔt\vec{F}=m\vec{a}\ \ \ \ \ \ \ \ \ \rightarrow\ \ \ \ \ \ \ \ \ \ \ \ \vec{F}=m\frac{\Delta v}{\Delta t}

Impulse

Impulse is defined as the change in the momentum of the object if a constant external force of F is applied to the object:


J⃗ =ΔP⃗  =F⃗Δt\boxed{\vec{J}\ =\Delta\vec{P}\ \ =\vec{F}\Delta t}


Wize Concept
If the net external force is zero, the change in momentum is zero and it remains conserved!

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Linear Momentum and Impulse


Linear Momentum

Linear Momentum (or simply Momentum) is defined as the product of mass and velocity of an object.
  • Momentum is a vector and measured in kg . m/s.

P⃗=mv⃗\boxed{\vec{P}=m\vec{v}}

If a force is applied to an object, the Second law of Newton could be written in terms of change in momentum as follows:

F⃗=ma⃗         →            F⃗=mΔvΔt\vec{F}=m\vec{a}\ \ \ \ \ \ \ \ \ \rightarrow\ \ \ \ \ \ \ \ \ \ \ \ \vec{F}=m\frac{\Delta v}{\Delta t}

F⃗=dp⃗dt\vec{F} = \frac{d\vec{p}}{dt}

Impulse

Impulse is defined as the change in the momentum of the object if a constant external force of F is applied to the object:


J⃗ =ΔP⃗  =F⃗Δt\boxed{\vec{J}\ =\Delta\vec{P}\ \ =\vec{F}\Delta t}


For a net external force which is changing in time, impulse is defined as below:


J⃗=ΔP⃗=∫titfF⃗(t)dt\boxed{\vec{J}=\Delta\vec{P}=\int_{t_i}^{t_f}\vec{F}\left(t\right)dt}


Exam Tip
If the plot of Force vs Time is given in the exam, impulse can be found as the area under the curve.

Wize Concept
Note that when the net external force is zero, momentum remains conserved!


Watch Out!
Note that a collection of objects with varying momenta can be described as one system with one center of mass velocity!

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Example: Impulse to a Soccer Ball


A soccer player applies an average force of 180 N during a kick. The kick accelerates a 0.45 kg soccer ball from rest to a speed of 18 m/s. What is the impulse imparted to the ball? What is collision time?

Solution:

We know that impulse is defined as:

J=Δp=pf−pi=FavgΔtJ = \Delta p = p_f-p_i=F_{avg}\Delta t

Since the ball is at rest before the kick, its initial momentum is zero.

pi=0p_i =0

So, we can easily calculate impulse as:

J=pf=mvfJ = p_f = mv_f
J=(0.45)(18)=8.1 kg.m/sJ = (0.45)(18) = 8.1\ kg .m/s

Furthermore, we can find the time duration of the kick again by looking at the definition of impulse:

J=FavgΔtJ = F_{avg} \Delta t

Δt=JFavg\Delta t =\dfrac{J}{F_{avg}}

Δt=8.1180=0.045 sec\Delta t = \dfrac{8.1}{180}= 0.045\ sec


A 2.5 kg Dodge ball hits the back wall of a school gym at 15 m/s at an angle of 50º to the wall. The ball is in contact with the wall for 0.15s before bouncing back at 12 m/s at an angle of 60º to the wall. What average force was exerted by the ball on the wall?


A force F=(2t2−5)NF=(2t^2-5) N is acting on a moving object with mass of 2kg in the direction of its motion for 5 seconds. If the initial speed of the particle is 5m/s.