a. A mass, m2, is initially at rest. A mass, m1, traveling with velocity, v1, collides with m2 in an elastic collision. After the collision, the two objects move with final velocities, v1f and v2f respectively. Using the laws of conservation of energy and momentum, show that the ratio of the final velocities in terms of m1 and m2 is given by: (v2f/v1f) = (2m1/m1 −m2) b. Linearize the expression in Exercise 1 by making the following substitutions: y≡m2/m1 andx≡v1f/v2f If y is plotted as a function of x, what values do you expect for the slope and intercept? c. Based on your result from Exercise2, what ratio of masses, m2/m1,results in final velocities, v1f andv2f that are equal in magnitude and opposite in direction?

Question

10-12-2022
Physics

contestada

a. A mass, m2, is initially at rest. A mass, m1, traveling with velocity, v1, collides with m2 in an elastic collision. After the collision, the two objects move with final velocities, v1f and v2f respectively. Using the laws of conservation of energy and momentum, show that the ratio of the final velocities in terms of m1 and m2 is given by: (v2f/v1f) = (2m1/m1 −m2) b. Linearize the expression in Exercise 1 by making the following substitutions: y≡m2/m1 andx≡v1f/v2f If y is plotted as a function of x, what values do you expect for the slope and intercept? c. Based on your result from Exercise2, what ratio of masses, m2/m1,results in final velocities, v1f andv2f that are equal in magnitude and opposite in direction?

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Conservation of kinetic energy, v2f/v1f = (2m1/m1 −m2),

ratio of masses, m2/m1 = 3.

By conservation of momentum,

m1v1 + 0 = m1v1f + m2v2f

=> v1 = v1f + (m2/m1)v2f

By conservation of kinetic energy,

(1/2)m1v1^2 + 0 = (1/2)m1v1f^2 + (1/2)m2v2f^2

m1v1^2 = m1v1f^2 + m2v2f^2

m1[v1f + (m2/m1)v2f]^2 = m1v1f^2 + m2v2f^2

m1[v1f^2 + (m2/m1)^2*v2f^2 + 2v1f.(m2/m1)v2f] = m1v1f^2 + m2v2f^2

m1v1f^2 + (m2^2/m1)v2f^2 + 2m2.v1f.v2f = m1v1f^2 + m2v2f^2

(m2^2/m1)v2f^2 + 2m2.v1f.v2f = m2v2f^2

(m2^2/m1)v2f + 2m2.v1f = m2.v2f

v2f[1 - m2/m1] = 2v1f

v2f/v1f = (2m1/m1 −m2)

b. From 3,

v1f/v2f = 1/2 - (1/2)(m2/m1)

Now y≡m2/m1 andx≡v1f/v2f

=> x = 1/2 - 1/2y

=> 2x = 1 - y

=> y = -2x + 1 ------------4

Comparing with y = mx + c,

Slope is m = -2 and intercept is c = 1

c. Now putting v1f = -v2f means x = -1

Therefore, from 4,

m2/m1 = y = -2*-1 + 1 = 3

=> m2/m1 = 3.

Every moving item possesses kinetic energy, which can be characterized as such. The energy that results from motion is simply referred to as kinetic energy. On the basis of the kind of motion that the objects are in, kinetic energy can be further divided into numerous sorts. For instance, rotational kinetic energy is the energy that a body with circular motion possesses, such as planets orbiting the sun; vibrational kinetic energy is the energy that an object has due to vibration, such as a vibrating phone; and translational kinetic energy is the energy that a body with motion possesses, such as a moving object from one location to another. Our daily lives make it simple to see translational kinetic energy in action.

Learn more about kinetic energy here:

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