Understanding Momentum
- Momentum (p) is defined as mass times velocity: (p = m \times v).
- Momentum is a vector quantity because velocity is a vector, while mass is scalar.
- Example: A 10 kg block moving at 6 m/s has a momentum of 60 kg·m/s.
For a deeper understanding of motion and momentum, see Understanding Motion: A Comprehensive Guide.
Calculating Impulse
- Impulse (J or I) is the product of force and the time interval during which the force acts: (J = F \times \Delta t).
- Example: Applying a force of 100 N for 8 seconds yields an impulse of 800 Ns.
- Impulse quantifies how much force is applied over a given duration.
The Impulse-Momentum Theorem
- Derived from Newton’s Second Law and acceleration definition:
- (F = m \times a = m \times \frac{\Delta v}{\Delta t})
- Multiplying both sides by (\Delta t) gives (F \times \Delta t = m \times \Delta v).
- This means Impulse equals the change in momentum.
- Essential formula for solving problems: (F \times \Delta t = m \times \Delta v).
Newton's laws are foundational to this theorem; explore detailed examples in Newton's Laws of Motion Explained with Real-Life Examples.
Force from Mass Flow Rate in Fluids
- Force can be calculated from the mass flow rate and velocity of a fluid:
- (F = \dot{m} \times v), where (\dot{m}) is mass flow rate.
- Example: Water flowing at 5 kg/s at 20 m/s produces a force of 100 N.
Force as the Rate of Change of Momentum
- For calculus-based analysis, force is the derivative of momentum with respect to time:
- (F(t) = \frac{dp}{dt}).
Conservation of Momentum in Collisions
Inelastic Collisions
- Objects collide and stick together, moving at a common final velocity.
- Momentum is conserved, but kinetic energy is not.
- Conservation Equation: (m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f).
Elastic Collisions
- Objects collide and bounce off each other.
- Both momentum and kinetic energy are conserved.
- Conservation of momentum: (m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2').
- Conservation of kinetic energy: (\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_1 v_1'^2 + \frac{1}{2} m_2 v_2'^2).
- Simplified velocity relation for elastic collisions: (v_1 + v_1' = v_2 + v_2').
- To solve for unknown velocities, use a system of equations based on momentum and energy conservation.
For more on energy aspects in collisions, refer to Conservation of Energy: Calculating Speeds and Heights in Physics Problems.
Summary of Key Equations
- Momentum: (p = m \times v)
- Impulse: (J = F \times \Delta t)
- Impulse-Momentum Theorem: (F \times \Delta t = m \times \Delta v)
- Force from fluid flow: (F = \dot{m} \times v)
- Force as time derivative of momentum: (F(t) = \frac{dp}{dt})
- Conservation of momentum for inelastic collision: (m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f)
- Conservation of momentum and energy for elastic collision
This comprehensive overview equips you with the critical formulas and concepts needed to master impulse, momentum, and collisions for your physics exam preparation. For worked examples and practice tests, refer to the linked resources.
in this video I'm going to do a quick review of impulse and momentum we're going to go over the key formulas that
you need to pass your next exam by the way for those of you who want example problems including a practice test on
this topic feel free to check out the links in the description section below so let's begin the first Formula you
need to be familiar with is this momentum is mass times velocity
as a vector you could describe it this way momentum is a vector mass is scalar but velocity is a
vector so let's say if you have a block that rests across a horizontal floor and let's say this is a 10 kgam block and
it's moving at a speed of 6 m/ second the momentum of this block is going to be 10 kg
that's the mass times the velocity of 6 m/s and so the momentum is going to be 60 with the units kilog time
m/s so that's how you can calculate momentum it's simply mass time velocity think of it as mass in
motion so make sure you write this down this is the first equation that you want to know for your
test now the next thing you need to know how to calculate is
impulse now some textbooks may use the letter J for impulse sometimes I've used J sometimes
I use I I for impulse but also I could represent inertia too so but impulse is equal to force multiplied by
time that's how you calculate impulse so imagine if you have a block and if you apply a
force of 100 Newtons for a time period of 8 seconds in previous chapters you've
learned about forces but in this chapter with impulse now we've Incorporated time with Force
how long is the force being applied to an object because that's important so here we have a force of 100
Newtons being applied on its object for 8 seconds the impulse imparted to this object is going
to be F delta T so it's 100 Newtons * 8 seconds which is 800 Newton
seconds so that's how you can calculate impulse impulse tells you how much force is being applied to an
object for a certain amount of time period so an Impulse of 800 can mean many
things it could be you're applying a force of 100 Newtons for 8 seconds or you're applying a large force of a th000
new for 8 seconds so it just tells you you know how much force you're applying for the
amount of time period combined now impulse and momentum they're related
impulse is equal to the change in momentum and here's how we can derive that so according to Newton's Second Law
force is equal to mass time acceleration and we know that the acceleration is the rate at which
velocity changes with respect to time acceleration is the change in velocity divided by the change in
time perhaps you remember this equation V final is equal to V initial plus a t if you solve for acceleration it will be
V final minus V initial over time in other words it's the change in velocity with respect to
time now if we multiply both sides by delta T we get that F delta T is equal to M Delta V on the left Force
multiplied by time gives us impulse on the right we know that masstimes velocity is momentum so mass times the
change in velocity is the change in momentum so this is related to the impulse momentum theorem now typically
the formula that you need to use when solving problems with this topic is this equation this is the form of the
equation that's going to be most helpful so that is the impulse momentum theorem so make sure you know these
three equations number one momentum is mass times velocity number two impulse is force multiplied
by time number three the impulse momentum theorem F delta T is equal to M Delta
V now going back to Newton's Second Law particularly this form there's a there's a lot of other equations that we
can get from this form so I'm going to adjust this
equation let's say if you have a person and he has a water hose in his hand and out of it he's going to shoot
out water let's say water is coming out of the hose at a speed of 20 m/
second so that's how fast it's coming out and then we need to talk about the quantity of water coming out there's
something called the mass flow rate so let's say water is leaving at a mass flow rate of 5 kg per
second this equation can help us to calculate the force that this fluid will exert on an
object let's say if you're blasting out water from the host on a block we can calculate the force exerted by
the water on this block so I'm going to move delta T to this position under M so we can
describe forces as being the mass flow rate Delta M over delta T time V so instead of the change
in velocity we have the change in mass in this case the velocity of water coming out of the Hol is is going to be
relatively constant so in this problem the mass flow rate is 5 kg per
second so every second 5 kog of water is coming out of that hose now the speed of the water is 20
m/s when we combine these two we get a force of 100 Newtons that's going to be the force exerted by the water on the
Block so that's another way in which you can calculate the force that a fluid exerts on an object this could be a
fluid like water it could be a fluid like air as long as you know the mass flow rate and the speed at which that
fluid is moving you can calculate the force I should really highlight this equation so this is the fourth equation
you want to know force is equal to mass flow rate multiplied by the velocity now for those of you who are taking
calculus with physics the next Formula is going to be more applicable to you so going back to this equation
instead of trying to get the mass flow rate from this equation we're going to focus on this
part we know that masstimes velocity is momentum so mass times the change in velocity is the change of momentum so
this is another way to describe force force can also be described as the rate at which momentum is
changing in other words force is the derivative of the momentum function with respect to
time so if you know the function for momentum you can get the function for force with respect to
time so F of T this is force as a function of time is the derivative of the momentum
function so these are some other ways in which you can calculate the force acting on an object if you know
the momentum function so I would write these equations number five and number six you
can add those to your notes now let's talk about the conservation of
momentum so let's say we have a horizontal frictionless floor and we have block
one let's say block one is moving at a speed of 2 m/s and it strikes block
two afterward the two blocks they stick together and they're moving at the same speed and you want to find out what this
final speed is here we have what is known as an inelastic collision whenever you have
two objects colliding if they stick together it's a collision but it's in elastic because you're going to have
loss of kinetic energy for elastic collisions both momentum and energy is conserved kinetic
energy but for inelastic collisions only momentum is conserved so this is an example of an inelastic collision
anytime they stick together and they don't bounce off you're dealing with an inelastic
collision momentum is conserved but kinetic energy is not conserved for for an in inelastic
collision so before the Collision we have the momentum of object one M1 V1 Plus momentum of object
two after the Collision we're going to have the momentum of both objects as well so this is the conservation of
momentum formula you want to make sure you write this down the basic idea behind this equation
is that the total momentum of the system before the Collision is equal to the total momentum of the system after the
Collision assuming no external forces are acting on a system because if there are external forces they will change the
total momentum of the system so this only holds true if there are no external forces acting on the
system so make sure you add this equation this is number seven now for this particular problem
because they stick together V1 final and V2 final they're the same they just equal V F so for this
particular situation you could shorten this equation to this form it's going to be M1 +
M2 times the final speed because they stick together so they move at the same final
speed so for any conservation of momentum problem you could use equation 7 but if they stick together at the end
the two objects after they Collide you could use equation 8 to solve it now let's say if we have a ball
moving at a speed of 6 m/ second and then strikes ball two which was initially moving at 2 m/ second and then
they bounce off each other ball one goes back in this direction let's say at 3 m/ second and you're trying to find out the
speed at which ball two uh moves in the to the right in other words you're trying to find V2
final a lot of times not always if they bounce off each other it could be an elastic
Collision but you want to make sure the problems specify that so if it's a perfectly elastic Collision that means
that kinetic energy is conserved as well as momentum so for this problem you would need to
use equation 7 to solve it so you have the conservation of momentum
equation now you're also let's say if you don't know this speed because if you know
V1 final you can probably just use this equation to get the answer but if you don't know V1 final you need another
equation to solve this so this is when you need to use conservation of energy that is the total initial energy is
equal to the total final energy so this would be K initial or K1 plus
ke2 equals K1 final plus K2 final now this form of the equation this going to be a lot of work a lot of
algebra if you use it but if you want to get the answer faster you could use the simplified equation that comes from this
equation and that equation is this which we'll describe as number 10 V1 plus V1 final is equal to V2 plus V2 final so if
you have an elastic Collision where you're missing the final speeds of both objects you need to use equation seven
and equation 8 to solve it so you need to write a system of equations to figure that problem
out if you're given V1 final but not V2 you could just use equation 7 but if you're missing two variables you have to
use two equations to solve two variables so you'll need equation 7 and 10 I do have example problems on this on my
other videos on YouTube so uh feel free to take a look at that when you get a chance and you might see this
also on my practice test too so that's basically it for this video those are the main equations you
need for this chapter on impulse and momentum as well as elastic collisions and in elastic collisions thanks for
watching
Momentum is the product of an object's mass and velocity (p = m × v) and represents the quantity of motion, while impulse is the product of a force applied over a time interval (J = F × Δt). Impulse measures the change in momentum caused by that force over the duration it acts, directly linking force and motion changes through the impulse-momentum theorem.
The impulse-momentum theorem states that impulse equals the change in momentum (F × Δt = m × Δv). To use it, calculate the impulse from the known force and time interval, then set it equal to the mass times the change in velocity to solve for unknown quantities like final velocity or force. This approach is useful for analyzing collisions or sudden force applications.
For inelastic collisions, where objects stick together, use momentum conservation: m₁v₁ + m₂v₂ = (m₁ + m₂)v_f. For elastic collisions, both momentum and kinetic energy are conserved, so solve the system: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂' and (1/2)m₁v₁² + (1/2)m₂v₂² = (1/2)m₁v₁'² + (1/2)m₂v₂'². This lets you find post-collision velocities accurately.
In fluid dynamics, force can be calculated as the product of mass flow rate ( m) and velocity (v), expressed as F = m × v. For example, if water flows at 5 kg/s with a velocity of 20 m/s, the force exerted by the flow is 100 N. This relationship helps analyze forces generated by moving fluids.
Momentum is a vector because it depends on velocity, which has both magnitude and direction, while mass is scalar. This means momentum has direction and must be treated using vector addition, especially when combining momenta from multiple objects or directions, crucial for accurate conservation analyses in collisions.
Force can be expressed as the time rate of change of momentum, F(t) = dp/dt. In calculus, by differentiating the momentum function with respect to time, you find the instantaneous force applied. This approach is vital when dealing with forces that vary over time rather than being constant.
Examples include a baseball bat hitting a ball, where the impulse applied changes the ball's momentum; car collisions demonstrating conservation of momentum; and a rocket propulsion system where mass flow rate and velocity produce thrust force. Understanding these helps relate physics concepts to everyday phenomena and engineering applications.
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