1. Physics 215 – Fall 2014Lecture 08-11 Welcome back to Physics 215 Today’s agenda: More on momentum, collisions Kinetic and potential energy Potential energy of a spring

2. Physics 215 – Fall 2014Lecture 08-12 Current homework assignment HW7: –Knight Textbook Ch.9: 54, 72 –Ch.10: 48, 68, 76 –Ch.11: 50, 64 –Due Wednesday, Oct. 22 nd in recitation

3. Physics 215 – Fall 2014Lecture 08-13 Momentum is a vector! Must conserve components of momentum simultaneously In 2 dimensions:

4. Physics 215 – Fall 2014Lecture 08-14 1.0 m/s 2.0.25 m/s 3.0.5 m/s 4.1 m/s A student is sitting on a low-friction cart and is holding a medicine ball. The student then throws the ball at an angle of 60° (measured from the horizontal) with a speed of 10 m/s. The mass of the student (with the car) is 80 kg. The mass of the ball is 4 kg. What is the final speed of the student (with car)?

5. Physics 215 – Fall 2014Lecture 08-15 At the intersection of Texas Avenue and University Drive, a blue, subcompact car with mass 950 kg traveling east on University collides with a maroon pickup truck with mass 1900 kg that is traveling north on Texas and ran a red light. The two vehicles stick together as a result of the collision and, after the collision, the wreckage is sliding at 16.0 m/s in the direction 24 o east of north. Calculate the speed of each vehicle before the collision. The collision occurs during a heavy rainstorm; you can ignore friction forces between the vehicles and the wet road.

6. Physics 215 – Fall 2014Lecture 08-16 Kinetic Energy For an object of mass m moving with speed v: K = (1/2)mv 2 Energy of motion scalar! Measured in Joules -- J Newton’s Laws are vector equations Sometimes more appropriate to consider scalar quantities related to speed and mass

7. Physics 215 – Fall 2014Lecture 08-17 Collisions If two objects collide and the net force exerted on the system (consisting of the two objects) is zero, the sum of their momenta is constant. The sum of their kinetic energies may or may not be constant.

8. Physics 215 – Fall 2014Lecture 08-18 Elastic and inelastic collisions If K is conserved – collision is said to be elastic e.g., cue balls on a pool table K A,i + K B,i K A,f + K B,f Otherwise termed inelastic e.g., lump of putty thrown against wall K A,i + K B,i K A,f + K B,f Extreme case = completely inelastic -- objects stick together after collision

9. Physics 215 – Fall 2014Lecture 08-19 Cart A moving to the right at speed v collides with an identical stationary cart (cart B) on a low-friction track. The collision is elastic (i.e., there is no loss of kinetic energy of the system). What is each cart’s velocity after colliding (considering velocities to the right as positive)?

10. Physics 215 – Fall 2014Lecture 08-110 Check conservation of momentum and energy

11. Physics 215 – Fall 2014Lecture 08-111 Elastic collision of two masses m1m1 v 1i Momentum  m 1 v 1i + 0 = m 1 v 1f + m 2 v 2f Energy  (1/2)m 1 v 1i 2 + 0 = (1/2)m 1 v 1f 2 + (1/2)m 2 v 2f 2 m2m2 m1m1 m2m2 v 2i = 0v 1f v 2f

12. Physics 215 – Fall 2014Lecture 08-112 Special cases: (i) m 1 = m 2 m1m1 v 1i m2m2 m1m1 m2m2 v 2i = 0v 1f v 2f

13. Physics 215 – Fall 2014Lecture 08-113 Special cases: (ii) m 1 << m 2 m1m1 v 1i v 1f v 2i = 0v 2f

14. Physics 215 – Fall 2014Lecture 08-114 Relative velocities for elastic collisions For example, velocity of m 2 relative to m 1 = v 2 - v 1 With m 2 stationary initially, v 2f - v 1f = v 1i In general, for elastic collisions, relative velocity has same magnitude before and after collision

15. Physics 215 – Fall 2014Lecture 08-115 Gravitational Potential Energy For an object of mass m near the surface of the earth: U g = mgh h is height above arbitrary reference line Measured in Joules -- J (like kinetic energy)

16. Physics 215 – Fall 2014Lecture 08-116 Total energy for object moving under gravity  E = U g + K = constant  * E is called the (mechanical) energy  * It is conserved: (½) mv 2 + mgh = constant

17. Physics 215 – Fall 2014Lecture 08-117 A ball of mass m=7 kg attached to a massless string of length R=3 m is released from the position shown in the figure below. (a) Find magnitude of velocity of the ball at the lowest point on its path. (b) Find the tension in the string at that point.

18. Physics 215 – Fall 2014Lecture 08-118 Stopped-pendulum demo Pendulum swings to same height on other side of vertical What if pendulum string is impeded ~1/2- way along its length? Will height on other side of vertical be: 1.Greater than original height 2.Same as original height 3.Less than original height?

19. Physics 215 – Fall 2014Lecture 08-119 A block is released from rest on a frictionless incline. The block travels to the bottom of the left incline and then moves up the right incline which is steeper than the left side.

20. Physics 215 – Fall 2014Lecture 08-120 Springs -- Elastic potential energy Force F = -kx (Hooke’s law) Area of triangle lying under straight line graph of F vs. x = (1/2)(+/-x)(-/+kx) F x F = -kx U s = (1/2)kx 2 frictionless table

21. Physics 215 – Fall 2014Lecture 08-121 (Horizontal) Spring frictionless table (1/2)kx 2 + (1/2)mv 2 = constant x = displacement from relaxed state of spring Elastic potential energy stored in spring: U s = (1/2)kx 2

22. Physics 215 – Fall 2014Lecture 08-122 A 0.5 kg mass is attached to a spring on a horizontal frictionless table. The mass is pulled to stretch the spring 5.0 cm and is released from rest. When the mass crosses the point at which the spring is not stretched, x = 0, its speed is 20 cm/s. If the experiment is repeated with a 10.0 cm initial stretch, what speed will the mass have when it crosses x = 0 ? 1.40 cm/s 2.0 cm/s 3.20 cm/s 4.10 cm/s

23. Physics 215 – Fall 2014Lecture 08-123 Mass hanging on spring Now oscillations are about equilibrium point of spring + mass Otherwise, motion is same as horizontal mass + spring on frictionless table (1/2)mv 2 = (1/2)ka 2 - (1/2)kz 2

24. Physics 215 – Fall 2014Lecture 08-124 Reading assignment Elastic energy Work Chapters 10 and 11 in textbook