Test – What to study The test will cover things we have done in class up to Lecture 9 Momentum will NOT be on the test Please look over old test (available online) Make sure you’re up on your reading ! Review error calculations (done in lab) Physics 1D03 - Lecture 25

Momentum Newton’s original “quantity of motion” a conserved quantity a vector Today: Newton’s Second Law in another form Momentum and Momentum Conservation Physics 1D03 - Lecture 25

Definition: The linear momentum p of a particle is its mass times its velocity: p  mv Momentum is a vector, since velocity is a vector. Units: kg m/s (no special name). (We say “linear” momentum to distinguish it from angular momentum, a different physical quantity.) Physics 1D03 - Lecture 25

Example 1 & 2 A car of mass 1500kg is moving with a velocity of 72km/h. What is its momentum ? Two cars, one of mass 1000kg is moving at 36km/h and the other of mass 1200kg is moving at 72km/h in the opposite direction. What is the momentum of the system ? Physics 1D03 - Lecture 25

Concept Quiz Which object will have the smallest momentum ? A 1.6x10-19kg particle moving at 1x103m/s A 1000kg car moving at 2m/s A 100kg person moving at 10m/s A 5000kg truck at rest Physics 1D03 - Lecture 25

Net external force = rate of change of momentum Newton’s Second Law If mass is constant, then the rate of change of (mv) is equal to m times the rate of change of v. We can rewrite Newton’s Second Law: or Net external force = rate of change of momentum This is how Newton wrote the Second Law. It remains true in cases where the mass is not constant. Physics 1D03 - Lecture 25

Example 3 v F Rain is falling vertically into an open railroad car which moves along a horizontal track at a constant speed. The engine must exert an extra force on the car as the water collects in it (the water is initially stationary, and must be brought up to the speed of the train). Calculate this extra force if: v = 20 m/s The water collects in the car at the rate of 6 kg per minute Physics 1D03 - Lecture 25

Solution v F Plan: The momentum of the car increases as it gains mass (water). Use Newton’s second law to find F. v is constant, and dm/dt is 6 kg/min or 0.1 kg/s (change to SI units!), so F = (0.1 kg/s) (20 m/s) = 2.0 N F and p are vectors; we get the horizontal force from the rate of increase of the horizontal component of momentum. Physics 1D03 - Lecture 25

The total momentum of a system of particles is the vector sum of the momenta of the individual particles: ptotal = p1 + p2 + ... = m1v1 + m2v2 + ... Since we are adding vectors, we can break this up into components so that: px,Tot = p1x + p2x + …. Etc. Physics 1D03 - Lecture 25

Example 4 A particle of mass 2kg is moving with a velocity of 5m/s and an angle of 45o to the horizontal. Determine the components of its momentum. Physics 1D03 - Lecture 25

Newton’s 3rd Law and Momentum Conservation Two particles interact: F12 F21 = -F12 m1 m2 Dp1= F12 Dt Dp2= F21 Dt Newton’s 3rd Law: F21 = -F12 The momentum changes are equal and opposite; the total momentum: p = p1 + p2=0 doesn’t change. The fine print: Only internal forces act. External forces would transfer momentum into or out of the system. eg: particles moving through a cloud of gas Physics 1D03 - Lecture 25

Conservation of momentum simply says that the initial and final momenta are equal: pi=pf Since momentum is a vector, we can also express it in terms of the components. These are independently conserved: pix=pfx piy=pfy piz=pfz Physics 1D03 - Lecture 25

Concept Quiz A subatomic particle may decay into two different particles. If the total momentum before is zero before the decay before, what is the total after ? 0 kg m/s depends on the masses depends on the final velocities depends on b) and c) together Before After Physics 1D03 - Lecture 25