Physic 231 Lecture 13 Main points of last lecture Work energy and nonconservative forces KE f KE0 PE f PE0 W nonconservative Power W P P Fv t Main points of today s lecture Impulses forces that last only a short time Momentum v v p mv vImpulse Momentum theorem v v v v F t p m v m v f v i Momentum conservation v v v Momentum and external forces ptot f p1 f p 2 f v v v p1 i p 2 i ptot i v v v Fext t ptot f p tot i Example In screeching to a halt a car leaves skid marks that are 65m long The coefficient of kinetic friction between the tires and the road is k 0 71 How fast was the car going before the driver applied the breaks x 65m KE f KE0 W friction k 0 71 KE0 W friction vf 0 Friction is in the opposite direction of the displacement W friction f k x 1 2 mv0 f k x k N x k mg x 2 v02 2 k g x v0 2 k g x 2 0 71 9 8 65 m s v0 30 m s Impulse forces that last a very short time There are many processes in which forces last a very short time and are difficult to mathematically describe Examples are Kicking striking batting dribbling a ball Various types of explosions firearms etc The typical time dependence of such impulse forces is described below 0 01s Momentum Typically we are interested in knowing how the velocity of an object is changed by an impulse force Since the impulse force is neither well understood mathematically nor reproducible it is not the natural quantity with which one describes such events Linear momentum and the change in linear momentum i e impulse are more useful for such descriptions The linear momentum of a particle of mass m is v v p mv The change in velocity is related to the change in momentum i e impulse v v v v v p p f pi m v f v i It is related to the average impulsive force v v v v f vi v v v v F t m a t m t m v f v i p t v v v F t v f vi m Bouncing balls Assuming each ball has the same mass which ball experiences the larger impulse a the first ball b the second ball Conceptual question You are a passenger in a car and not wearing your seat belt Without increasing or decreasing its speed the car makes a sharp left turn and you find yourself colliding with the right hand door Which is the correct analysis of the situation a Before and after the collision there is a rightward force pushing you into the door b Starting at the time of collision the door exerts a leftward force on you c both of the above d neither of the above Example A 0 4 kg ball is dropped from rest at a point 1 5 m above the floor The ball rebounds straight upward to a height of 0 8m What is the magnitude and direction of the impulse applied to the ball by the floor If the ball is in contact with the floor for 0 01 seconds what is the impulse force h 1 5 m 0 v r v v v p p f p0 m v f v0 hf 0 8 m m 0 4 kg 1 2 mv0 mgh0 v0 2 gh0 2 9 8 1 5 m s 5 4m s 2 1 2 mv f mgh f v f 2 gh f 2 9 8 0 8 m s 3 96m s 2 v v v p m v f v0 0 4kg 3 96m s 5 4m s 3 75kg m s upwards F p 3 75kg m s 375 N upwards t 0 01s Quiz Jack swings at a 0 2 kg ball that is moving west with a velocity of 40 m s and hits a line drive The leaves his bat with a velocity of 40 m s due east Assuming the ball is in contact with the bat for 0 010 s what is the average impulse force of the bat on the ball a 800N east m 0 2 kg b 1600 N east t 0 01 s c 1600 N west v0 40 m s d 800 N west vf 40 m s p 0 2kg 40m s 40m s 16kg m s east F p 16kg m s 1600 N east t 0 01s Example A dump truck is being filled with sand The sand falls straight downward from rest from a height of 2 00 m above the truck bed and the mass of sand that hits the trick per second is 55 0 kg s The truck is parked on the platform of a weight scale By how much does the scale reading exceed the combined weight of truck and sand Force on the sand Fs is given by Fs t ms vsand f vsand 0 ms vsand 0 ms 55kg s t Fs 55kg s vsand 0 1 ms v 2sand 0 ms gh v 2sand 0 2 gh 2 vsand 0 2 gh 2 9 8m s 2 2m 6 26m s Fs 55kg s 6 26m s 344 N upwards By Newton s third law the force on the truck is Ftruck 344 N downward This is the amount by which the weight of truck is increased Conservation of linear momentum Consider the collision of objects that interact with each other but whose interactions with the rest of the world can be neglected As an example one can consider two hockey pucks one larger and the other smaller that are sliding without friction on a frictionless ice surface From Newton s 3d law v v F21 F12 at all times On the average v v v v F21 F12 F21 t F12 t v v v v v v p 2 p1 p 2 f p 2 o p1 o p1 f If we reorganize terms v v v v p 2 f p1 f p 2 o p1 o v v ptot f ptot o Thus total momentum is conserved in isolated system i e one without external forces When there are external forces such as gravity v v v Fext t ptot f p tot i
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