Lecture 7 Collisions Durin the preious lecture we stared our discussion of collisions As it was stated last tie a collision is an isolated eent in which two or ore odies (the collidin odies) exert relatiely stron forces on each other for a relatiely short tie We hae also sudiided collisions into two ain types f kinetic enery of the syste of two collidin odies is unchaned durin the collision we called this collision the elastic collision f kinetic enery of the syste is not consered durin the collision we called this type of collision inelastic collision Reardless of what happens to the total enery of the syste the total linear oentu of the closed syste cannot chane The oentu for each of the collidin odies can chane ut the total oentu of the closed syste for any type of the collision should stay the sae nelastic collisions Let us start y considerin inelastic collision in one diension This collision takes place etween the two odies of asses and They oe alon the sae axis x efore as well as after the collision Let these odies hae elocities i and i efore the collision and f and f after the collision Since conseration of the linear oentu takes place for any type of collision we hae Pi Pf p p p p i i f f (7) i i f f where Pi pi p i is the oriinal oentu of the syste and Pf p f p f is the final oentu of the syste f one of the odies (the taret) was oriinally at rest ( i 0 ) and the collision was a copletely inelastic collision then odies will stick toether after the collision so they will hae the sae elocity V Then equation 7 ecoes V V i V i f f f we know the oriinal elocity of the first ody then the final elocity will e (7) V i (73)
which eans that the final elocity will e less than the oriinal elocity Since we are considerin a closed syste there are no external forces actin on it This eans that elocity for the center of ass of this syste is not chanin durin the collision and it is P p p p p co M i i f f Exercise 7 Ballistic pendulu (74) The allistic pendulu was used in older days to easure elocity of ullets This pendulu is a assie lock of ass M suspended on two ropes The ullet of ass is fired into that lock As a result the lock oes and rises for certain heiht h until it stops This heiht can e easured experientally Based on this inforation how one can find elocity of the ullet? la Please think aout this prole since you will e doin it as an experient in the Hint: The collision etween the lock and the ullet is an exaple of copletely inelastic collision since the lock and the ullet stick toether after the collision You can use conseration of the linear oentu durin this process Also note that conseration of enery works for this syste after the collision ut not durin the collision Elastic collisions Een thouh we hae already discussed that ost part of the real collisions are not elastic since soe part of kinetic enery is always lost ut in any cases these losses are not sinificant so we can approxiately treat collisions as elastic This eans that kinetic enery of each of the collidin odies ay chane ut the total kinetic enery of the syste will not chane durin the collision Aain let us consider one-diensional case first The est exaple will e the two illiard alls traelin alon the sae straiht line The first all of ass is oin with elocity i towards the second all of ass which is oriinally stationary i 0 Since the linear oentu is consered durin the collision we hae (75) i f f Since this is elastic collision the kinetic enery is also consered i f f (76)
Usually asses of the odies are known as well as the oriinal elocity of the first ody is known while the final elocities are not known To find these two unknowns one can use two equations 75 76 The first equation can e rewritten as d i i f f the second equation as d id i i f i f f So this ecoes or i f i f f f i f f f i i i i (77) This equation shows that f is in the sae direction as i so the taret will always e oin in the sae direction in which it was hit y the all Howeer the sin of f depends on the relation etween the asses the all can either oe forward or reound There are soe interestin special cases ) f oth asses are equal then f 0 and f i This predicts that head-on collision of odies with equal asses results in the exchane of their elocities The first ody will stop while the second ody will oe with the sae elocity as the oriinal elocity of the first ody This would e the ost desirale case for the illiard player ) A assie taret this eans that f i and f i The first ody siply ounces ack alon its incoin path with alost the sae speed The second ody oes forward at extreely law speed
3) f the second ody is ery liht copared to the first ody then and f i f i The first ody will continue with alost the sae speed as efore the collision while the second ody will oe een faster Exaple 7 Let us consider two etal spheres (pendulus) of asses and suspended on ertical cords oriinally just touchin each other Then one of the pendulus is pulled out raised to heiht h and released After that this pendulu underoes an elastic collision with the second pendulu which then rises for a axiu heiht h What is this heiht h? As the exaple says the collision is elastic External forces of raity and tension in the cords do not affect it since they are perpendicular to the spheres' elocities in the lowest point where this collision occurs Both elocities are directed alon the sae horizontal direction so we can apply equation 77 which shows that elocity of the second sphere after the collision is f i where i is the elocity of the first sphere just efore the collision We can also apply the enery conseration for each of the spheres efore and after the collision They for the closed syste with the earth since tension in the cord is always perpendicular to elocity and does not do any work or the first sphere we hae h h i i h i or the second sphere conseration of enery ies f f h h f h Coinin all this we hae
h h h h h h t is easy to see that if oth spheres hae the sae ass the second sphere will jup at the sae heiht fro which the first sphere was released Now let us see what will happen if the taret was not stationary ut oin The laws of conseration will ie (78) i i f f for oentu and i i f f (79) for kinetic enery We shall apply the sae ethod to sole these equations They can e rewritten as d i d i d id i d id i i f i f i f i f i f i f After soe aleraic work we will arrie to f i i f i i (70) (7) Exercise: Derie equations 7 fro 70 The last equation is copletely syetric to the exchane of the two odies f the collision is not head-on the odies will not oe alon the sae line and the prole will ecoe two-diensional prole But all of the equations are still workin except we hae to reeer that oentu of the syste is in fact a ector so (7) i i f f Equation 79 for enery which is scalar equation stays exactly as it was efore Equation 7 ies us two equations for the coponents of ectors
Let us consider the stationary taret n this case we can always choose the positie x direction to e the sae as the direction of the oriinal elocity of the first ody f after the collision the first ody akes anle and the second ody akes anle with the positie x direction then equation 7 ecoes ery siple t ies cos cos i f f 0 sin sin f f (73) Exaple 7 Two spherical particles of asses and are oin alon the perpendicular lines The first particle oes with elocity in the positie x direction The second particle oes with elocity in the positie y direction They collide at soe point and stick toether What is their elocity after the collision? This is an exaple of inelastic collision since the particles will stick toether The conseration of oentu for this collision ies us This equation is a ector equation since we hae two-diensional collision here So we hae to represent it in ters of coponents for coordinate axes Takin into account that the first particle has only x coponent of the elocity efore the collision and the second particle has only y coponent we arrie to This eans / x / y x y Accordin to Pythaorean Theore the final speed will e x y which is the anitude of the final elocity To find its direction let us calculate the tanent of the anle which it akes with positie x direction y tan x