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Topics for Today

Get a comprehensive understanding of linear momentum, collision, and impulse with this lab workbook. Available in the book store now!

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Topics for Today

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  1. Topics for Today • Lab workbooks are available in the book store • Set clicker to channel 44 • Linear momentum (9-3) • Collision and impulse (9-4) • Conservation of linear momentum (9-5) • Momentum and kinetic energy in collisions (9-6)

  2. Linear Momentum • Physicists really like conservation laws. Velocities of particles are not conserved, but a related quantity called the “momentum” is conserved. • Linear momentum is • Newton’s second law is then • If we insert the definition of momentum, we get back the 2nd law • Newton originally wrote his second law in the momentum form.

  3. Linear Momentum for a System • For a system of particles, the total linear momentum is • If you compare this to the definition of the center of mass, you find (after taking one time derivative) that where M is the total mass of the system. • If we take the time derivative, then we find 2nd law for a system • As before, this looks exactly like the 2nd law for a single particle.

  4. Collisions and Impulse • Sometimes, particles interact so fast that we can’t keep up with the details. In these cases, like collisions, we might not be able to find the force as a function of time, but we can find the integral of the force that we call the “impulse”. Integrating the 2nd law w.r.t. time, we find • Quiz – a baseball with mass of 150 grams and a horizontal speed of 40 m/s is hit with a bat. The ball ends up moving horizontally in the opposite direction with a speed of 60 m/s. What impulse did the bat give to the ball? A=15 kg, B=15 m/s, C=15 kg∙m/s, D=15 kg∙m/s2

  5. Collisions and Impulse • Usually, the force is some complicated function of time. • In some cases, we know only the average force, , exerted over some interval of time, . The impulse is then • Quiz – The baseball in the previous problem received an impulse of 15 kg∙m/s. If the bat and baseball were in contact for 0.7 milliseconds, what was the average force exerted on the bat by the ball? A=21 N, B=210 N, C=2100 N, D=21000 N

  6. Collisions and Impulse • What is the sign of Δpx? A = positive, B = negative, C = zero • What is the sign of Δpy? A = positive, B = negative, C = zero • What is the direction of ? A = +x, B = -x, C = +y, D = -y

  7. Conservation of Linear Momentum • If we have an isolated system with no external forces acting on it, then or or • Momentum is conserved! • Do clacker demo. Why is the number of balls moving conserved? • Do demo 1N20.20 Conservation of linear momentum – Air track. • Do demo with equal masses. • Quiz – what is the ratio of speeds (in the final state) if the ratio of masses is 1:3? A=1:1, B=1:3, C=3:3, D=3:1 • Do demo with 1:3 mass ratio.

  8. Conservation of Linear Momentum Quiz – a car accelerates from rest. In doing so the car’s momentum changes by a certain amount. The Earth’s moment changes by • Zero • The same amount • And equal and opposite amount • The answer depends on the interaction between the two.

  9. Momentum and Energy in Collisions • If two (or more) objects collide then , momentum is conserved. This is always true as long as there are no external forces. • While energy is also always conserved, this is not necessarily true of kinetic energy. We find two types of collisions: • Elastic collisions = kinetic energy is conserved • Inelastic collisions = kinetic energy is not conserved. • The greatest loss in kinetic energy occurs when two objects collide and then stick together. This is a “completely inelastic collision”.

  10. Inelastic Collisions • If two (or more) objects collide then , momentum is conserved. This is always true as long as there are no external forces. • While energy is also always conserved, this is not necessarily true of kinetic energy. We find two types of collisions: • Elastic collisions = kinetic energy is conserved • Inelastic collisions = kinetic energy is not conserved. • The greatest loss in kinetic energy occurs when two objects collide and then stick together. This is a “completely inelastic collision”.

  11. Inelastic Collisions • For inelastic collissions, we cannot use energy conservation, so we used momentum conservation, . • However, this doesn’t always lead to a unique answer. • For completely inelastic collisions, momentum conservation does give a unique answer • Thus, we can find the velocity of the objects once they have stuck together (and this works for any number of initial objects).

  12. Inelastic Collisions • Quiz – a Prius (mass = 1300 kg) and a Chevy Suburban (mass = 2600 kg) are traveling towards each other at 120 km/hr. They collide and stick together. What is the speed of the resulting wreck? • 20 km/hr • 40 km/hr • 60 km/hr • 80 km/hr • About 2×106 J of energy is lost in the collision. Where does it go? • Do inelastic collision on air table. →

  13. Inelastic Collisions Quiz – suppose rain fall vertically into an open cart rolling along a straight horizontal track with negligible friction. As a result of the accumulating water, the speed of the cart • increases • does not change • decreases

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