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Linear Momentum & Impulse

Define Linear Momentum = product of objects mass x velocity
A measure of how hard it is to stop an object. It is like a quantity of motion. How is it different from inertia?

Momentum (p) depends on: mass & velocity of object.
p = mv m in kg v in m/s Units are … kg m no name. s

Momentum is a Vector Quantity
Same direction as velocity All Energy KE too is a scalar

Change in momentum occurs any time an object changes velocity (speed or direction).

Momentum Change & Newton’s 2nd Law
F = ma F = m(Dv/Dt) FDt =mDv m (vf - vi) for const mass. FDt = Dp Impulse direction is same as F. Dp = Change in momentum

Equations of Momentum Change
Impulse J = change momentum. J =FDt = Dp pf – pi. Dp = mvf – mvi for velocity change with constant mass can factor out mass you can write, m (vf - vi) or mDv.

Increased force & contact time on object give greatest Dp = mDv.

The more time in contact, the less force needed to change p.

The quantity FDt (or Ft) is called impulse (J).
Impulse (J) is the momentum change. It has the same units. kg m or Ns s

Time of impact Force Impulse Dp Acceleration Damage done.
1. A bus driving east hits a mosquito flying west. Compare the impacts of each on the bus and the bug: Time of impact Force Impulse Dp Acceleration Damage done.

Changing momentum: bringing objects to rest with impulse.
Catch the egg without breaking vs dropping on ground. Fall from building onto cement vs. airbag. Same impulse, more time = less force.

Dp = m Dv. = 1 kg( - 8 m/s – 6 m/s) - 14 Ns
2. Find the change in momentum of a 1 kg mass which is dropped and hits the floor with a velocity of 8 m/s. It bounces back up with 6 m/s. Dp = m Dv. = 1 kg( - 8 m/s – 6 m/s) - 14 Ns

Stand on a skateboard catch a ball and bring it to rest or let it bounce off?
Bouncing causes bigger impulse than absorbing or giving with the motion.

Graphs

Constant force F - t graph: Dp /Impulse is area under curve FDt.
Force N

3. Non-Constant Force Force vs. time graph
3. Non-Constant Force Force vs. time graph. The area under the curve = impulse or Dp change in momentum. How much impulse is each box on the graph? 5 Ns.

4. What is the change in velocity imparted to the 0.8 kg object below?

IB Style Question

5. Water is poured from 0. 5 m onto a pan balance at 30 L/min
5. Water is poured from 0.5 m onto a pan balance at 30 L/min. Assume vf of water = rWat = 1 kg/L. 1. Estimate the velocity of the water upon hitting the pan. (Assume the stream starts from rest). 2. Estimate the mass of water hitting the pan each second. 3. Assuming the water’s velocity after hitting the pan goes to zero, estimate the reading on the pan balance in grams.

v2 = 2as. v2 = 2(10)(0.5) = v = 3.2 m/s Mass water/sec, 30 L / 60 s x 1 kg/ L = 0.5 kg/sec so in 1 second 0.5 kg mass arrives at the pan balance. Water changes momentum FDt = mDv. The force on the balance = mDv/t, (0.5kg)(3.2 m/s)/ 1 s = 1.6 N = 160 grams.

Hwk Kerr. Pg 72 # 6-7

Newton’s First Law Object at rest or constant velocity has not Fnet. Upward = Downward.

Newton’s 3rd Law Object A exerts Force F, on object B, then object B exerts equal but opposite force on A. F a,b = - F b,a.

Conservation Momentum particle interaction N3
FAB = - FBA. FDt = mDv mDva = - mDvb. t t Contact time, t, is the same they cancel. m (vfa – via ) = - m (vfb – vib ) Expand and rearrange, collect vi on one side, vf on the other. S pi = Spf (Conservation of momentum).

Conservation of Momentum
If no external force acts on a closed system, the total momentum within the system remains unchanged even if objects interact. Momentum can be transferred between objects.

What is a system? Two or more objects that interact in motion. One may transfer part or all of its momentum to the other(s). Common examples: collisions, explosions.

6. Bounce a ball off the floor
Did the momentum of the ball change? Was conservation of momentum obeyed?

What happened to the momentum?
How much momentum was gained by Earth? The ball’s mass is 0.25-kg. It’s initial speed was 5.0 m/s, and its final speed was 3.0 m/s. What was the change in velocity of Earth due to the collision? (mass Earth = 6.0 x 1024 kg.) The impulse on the ball: 0.25 (8 m/s) = 2.0 Ns. 2.0 Ns = mDv Dv = 2 Ns / 6 x 1024 kg

SPbefore = Spafter m1v1 + m2v2 = m1fv1f + m2fv2f
To Calculate: SPbefore = Spafter m1v1 + m2v2 = m1fv1f + m2fv2f v1 and v2 velocities for objects one and two. m1 and m2 masses of objects

One Ball transfers all its momentum.

Conservation of Momentum Calc’s
Total momentum before = total after interactions. The direction of the total momentum is conserved as well. Collisions. Explosions Pushing apart.

Elastic & Inelastic Collisions
Elastic: no KE (velocity) lost (to heat, light, sound etc.) Usu. Involves objects that don’t make contact. KE before = KE aft. Inelastic: involves greatest loss of KE (velocity). Often objects stick together.

Recoil: objects initially at rest explode or push apart

Recoil illustrates conservation of momentum where initial and final momentum = 0. 0 = p1 + p2.

7. On July 4th my family likes to shoot off fireworks
7. On July 4th my family likes to shoot off fireworks. One rocket was shot straight up, climbed to a height 18-m and exploded into hundreds of pieces in all directions at its highest point. Thinking about conservation laws, think about the rocket at its highest point just before & just after it explodes: How does the rocket’s momentum compare before & after the explosion? How does its KE compare before & after the explosion?

Throw a ball off the wall.
How is momentum conserved? What is the system?

Systems, External & Internal Force
If system is single astronaut, then external force applied by astronaut 2, momentum not conserved –it changes. If system is 2 astronauts, then the force is internal and total momentum is conserved.

State Newton 3 If 2 objects interact, the force exerted by on object A by object B (Fa,b), is equal in magnitude but opposite in direction to the force exerted on object B by object A, (-Fb,a).

1. A lamp of weight W is suspended by a wire fixed to the ceiling
1. A lamp of weight W is suspended by a wire fixed to the ceiling. With reference to Newton’s third law of motion, the force that is equal and opposite to W is the: A. tension in the wire. B. force applied by the ceiling. C. force exerted by the lamp on the Earth. D. force exerted by the Earth on the lamp

2. A student is sitting on a chair
2. A student is sitting on a chair. One force that is acting on the student is the pull of gravity. According to Newton’s third law, there must be another force which is: A. the upward push of the chair on the student. B. the downward force on the student. C. the downward push of the chair on Earth. D. the upward force on Earth.

3. What is the reaction force for the following:
A 0.5 kg bird glides above the earth’s surface. It’s wings push down on the air with its weight, 5-N, so:

How can anything have Fnet and accelerate?
Acceleration is caused by the Fnet on a single object. It is the sum of all the forces. Action/Reaction occurs for different objects.

Hwk in Kerr pg 72 # 8 – 9 Show work. IB set momentum .

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