7) A one-dimensional elastic collision occurs between two masses on a frictionle

Question

7) A one-dimensional elastic collision occurs between two masses on a frictionless surfaces. The initial kinetic energies of the masses were 9 J and 6 J. After the collision the kinetic energy of one of the masses is 17 J. What is the kinetic energy of the other mass? 8) There is a ramp which connects two horizontal surfaces. The upper level surface is 7.0 m above the lower level surface. The closest edges of the level surfaces are 11 m apart. The transition between the horizontal surfaces and the ramp are smoothed to avoid any additional forces messing up the ride. At the end of the lower surface (not the end touching the ramp) is a massless (of course) spring with spring constant = 45 N/m. The distance between the ramp and the start of the spring is 5 m. All of the surfaces are frictionless. There is a 3 kg box (Box A) at the top of the ramp, barely not moving. On the lower surface between the edge of the ramp and spring is a second box (Box B), mass 7 kg, initially at rest. The top box is lightly touched and it begins to descend (initial speed is close enough to 0 m/s). Box A will elastically collide with Box B. Box B will then slide into the spring and stick to it. Box A will do something which the reader will determine as a problem. Assume that the sizes of the boxes are big enough to collide but not big enough to consider the dimensions of the boxes as significant. All speeds and velocities should be relative to surfaces and ramp. a) How fast is Box A going just before it collides with Box B? b) What are the velocities of Boxes A and B just after the collision? c) How far up the ramp will Box A go after the collision? [If Box A makes it to the upper level or if Box A does not move toward the incline after the collision, then show it mathematically.] d) After Box B sticks to the spring, what is the maximum compression of the spring? e) After Box B sticks to the spring, what is the period of oscillation? f After Box B sticks to the spring, what is the formula to describe motion, x(t), of the spring? 9) A8 kg mass is moving at +5 m/s along a frictionless floor toward a 12 kg mass at rest. What will be the velocities of the masses after the collision, if the collision is a) perfectly inelastic (i.e. they stick together)? b) perfectly elastic?

Explanation / Answer

7.

By Conservation of emergy

KE1,i+KE2,i =KE1,f +KE2,f

9+6=17+KE2,f

KE2,f =15-17 = - 2 Joules

9.

a)

For perfectly inelastic collision

BY Conservtion of momentum

M1V1+M2V2 =(M1+M2)V

8*5+12*0 =(8+12)*V

V=40/20 =2 m/s

V1f=V2f=2 m/s

b)

For perfectly elastic collision

M1V1i+M2V2i =M1V1f +M2V2f

8*5+0=8V1f +12V2f

V2f =(40-8V1f)/12

By conservation of energy

(1/2)M1V1i2 +(1/2)M2V2i2 =(1/2)M1V1f2 +(1/2)M2V2f2

8*52+0 = 8*V1f2 +12*[(40-8V1f)/12]2

200 =8V1f2 +(1600-640V1f+64V1f2)/12

2400 = 96V1f2 +1600-640V1f+64V1f2

160V1f2-640V1f-800=0

V1f2-4V1f-5=0

V1f=5 m/s or -1 m/s

V1f =-1 m/s

V2f=(40-8(-1))/12=4 m/s

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