A 4.5 kg object moving in the +x direction at 5.5 m/s collides head-on with a 2.

Question

A 4.5 kg object moving in the +x direction at 5.5 m/s collides head-on with a 2.7 kg object moving in the -x direction at 4.0 m/s. Find the final velocity of each mass for each of the following situations. (Take the positive direction to be +x.)
(a) The bodies stick together.
m/s (4.5 kg mass)
m/s (2.7 kg mass)

(b) The collision is elastic.
m/s (4.5 kg mass)
m/s (2.7 kg mass)

(c) The 4.5 kg body is at rest after the collision.
m/s (4.5 kg mass)
m/s (2.7 kg mass)

(d) The 2.7 kg body is at rest after the collision.
m/s (4.5 kg mass)
m/s (2.7 kg mass)

(e) The 4.5 kg body has a velocity of 4.0 m/s in the -x direction after the collision.
m/s (4.5 kg mass)
m/s(2.7 kg mass)

Are the results in (c), (d), and (e) "reasonable"? Explain.

Explanation / Answer

since no external work are done so we can apply linear momentum conservation law

(a) when body stick to each other
(4.5*5.5-2.7*4)=(4.5+2.7)*V
so V = 1.9375 m/s

(b) the collision is elastic, this means energy will also conserve
(4.5*5.5-2.7*4) = (4.5*x + 2.7*y) eq.1 .. (momentum conservation equation)
where x and y are the velocity of the body 4.5 kg and 2.7 kg respectively.

(4.5*5.52/2)+(2.7*42/2) = {(4.5*x2/2) + (2.7*y2/2)} eq.2 ...(energy conservation equation)

on solving eq 1 and eq 2 we will get,

x (4.5 kg mass)= -2.2767 m/s = 2.2767 m/s in -ve x-direction

y(2.7 kg mass)= 8.9611 m/s

(c) The 4.5 kg body is at rest after the collision.

linear momentum conservation law

(4.5*5.5-2.7*4)=(2.7*V+4.5*0)

V = 5.167 m/s

(d) The 2.7 kg body is at rest after the collision.

linear momentum conservation law

(4.5*5.5-2.7*4)=(2.7*0+4.5*V)

V = 3.1 m/s

(e) The 4.5 kg body has a velocity of 4.0 m/s in the -x direction after the collision.

linear momentum conservation law

(4.5*5.5-2.7*4)=(2.7*V-4.5*4)

V = 31.95 m/s

(c) is reasonable but (d) and (e) can not.

because 4.5 kg body have initial momentum 24.75 kgm/s whereas 2.7 kg body has 10.8 kgm/s.

in (c) 4.5 kg body can give their whole momentum to 2.7 kg body and can be in rest to make 2.7 kg body moving with 5.167 m/s.

whereas in (d) when 2.7 kg became in rest its giving his whole momentum which is only 10.8 kgm/s which is not more than 24.75 kgm/s. so can not overcome it.

similary in case (e) for making the body 4.5 kg in moving with 4 m/s in -ve x-direction it needs 42.75 kgm/s momentum which is more than 2.7 kg body momentum, so it can't be provided by body 2.7 kg.

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