Two bricks with masses m1=2.5 kg and m2=4 kg can slide without friction on a hor

Question

Two bricks with masses m1=2.5 kg and m2=4 kg can slide without friction on a horizontal linear track. m1 is moving with a velocity v1i=5 m/s when it collides with m2 which is initially at rest. (

a) Calculate the velocity of m1 assuming mechanical energy is conserved in the collision.

(b) We now drop the assumption of the collision being elastic and consider the general case of a collision where linear momentum is conserved. Determine the largest possible amount of mechanical energy that can be converted into thermal energy during this collision and the associated velocities of the bricks after the collision.

(c) After the collision m2 actually ends up moving a velocity v2f=2 m/s. How much mechanical energy was converted into thermal energy during the collision?

Explanation / Answer

m1 = 2.5 kg
m2 = 4 kg

v1i = 5 m/s
v2i = 0

(a)
Using Momentum Conservation -
m1 * v1i + m2 *v2i = m1 * v1f + m2 * v2f
2.5 * 5 + 0 = 2.5 * v1f + 4 * v2f
12.5 =  2.5 * v1f + 4 * v2f --------------1

Initial Kinetic Energy = 0.5 *m1v1i^2
Final Kinetic Energy = 0.5*m1v1f^2 + 0.5*m2v2f^2

Initial Kinetic Energy = Final Kinetic Energy
0.5 *mv1i^2 =  0.5*mv1f^2 + 0.5*mv2f^2
2.5*5^2 = 2.5*v1f^2 + 4 * v2f^2 -----------------2

Using eq 1 and 2 Solving for v1f & v2f -
v1f = -1.15 m/s
V2f = 3.85 m/s
Velcoity of m1 = -1.15 m/s. (-ve sign means it started moving in the opposite direction just after collision.)

(b)
For Maximum Conversion of Energy into Thermal Energy -
A perfectly inelastic collision occurs when the maximum amount of kinetic energy of a system is lost.

Let the final Velocity of m1 and m2 = vf

Using Momentum Conservation -
m1 * v1i + m2 *v2i = (m1 + m2) * vf
2.5 * 5 + 0 = (2.5 + 4) * vf
vf = 1.92 m/s --------------1

Thermal Energy = Inital Kinetic Energy - Final Kinetic Energy
Thermal Energy =  0.5 *m1v1i^2 - 0.5*m1vf^2 - 0.5*m2vf^2
Thermal Energy =  0.5 *2.5*5^2 - 0.5*2.5*1.92^2 - 0.5*4*1.92^2
Thermal Energy = 19.27 J

Maximum amount of Mechanical energy converted into thermal energy during the collision, = 19.27 J

(c)
Using Momentum Conservation -
m1 * v1i + m2 *v2i = m1 * v1f + m2 * v2f
2.5 * 5 + 0 = 2.5 * v1f + 4 * 2
v1f = 1.8 m/s --------------1

Initial Kinetic Energy = 0.5 *m1v1i^2
Final Kinetic Energy = 0.5*m1v1f^2 + 0.5*m2v2f^2

Thermal Energy = Inital Kinetic Energy - Final Kinetic Energy
Thermal Energy =  0.5 *m1v1i^2 - 0.5*m1v1f^2 - 0.5*m2v2f^2
Thermal Energy =   0.5 * 2.5 * 5^2 - 0.5 * 2.5 * 1.8^2 - 0.5 * 4*2^2
Thermal Energy = 19.2 J

Mechanical energy converted into thermal energy during the collision, = 19.2 J

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