A railroad freight car of mass 3.30 104 kg collides with a stationary caboose car. They couple together, and 16.0% of the initial kinetic energy is transferred to thermal energy, sound vibrations, and so on. Find the mass of the caboose.
Law of momentum applies:
Mf*Vi=(Mf+Mc)Vf and given is mass Mf.
Law of eneryg applies
1/2 Mf(Vi)^2*.84=1/2(Mf+Mc)Vf^2
In equation 1), solve for Vf
Vf=vi(Mf)/(Mf+Mc)
Put that in to the second equation.
1/2 Mf(vi)^2(.84)=1/2 (Mf+Mc)(vi*Mf/(Mf+Mc))^2
.84=Mf/(Mf+mc)
and solve for mc.
To find the mass of the caboose, we can use the principle of conservation of energy. According to the problem statement, 16.0% of the initial kinetic energy is transferred to thermal energy and sound vibrations. Therefore, the remaining energy is conserved and converted to kinetic energy of the coupled system of the freight car and the caboose.
The initial kinetic energy before the collision is given by the formula:
KE_initial = (1/2) * m1 * v1^2,
where m1 is the mass of the freight car and v1 is its velocity.
After the collision, the initial kinetic energy is transferred to the coupled system, and we can express it as:
KE_final = (1/2) * (m1 + m2) * v2^2,
where m2 is the mass of the caboose and v2 is the final velocity of the coupled system.
From the problem statement, we know that 16.0% of the initial kinetic energy is lost, so the remaining energy is:
KE_final = (1 - 0.16) * KE_initial.
Setting these two equations equal to each other, we have:
(1/2) * (m1 + m2) * v2^2 = (1 - 0.16) * (1/2) * m1 * v1^2.
Canceling out the common factors, we get:
(m1 + m2) * v2^2 = 0.84 * m1 * v1^2.
Simplifying, we have:
v2^2 = (0.84 * m1 * v1^2) / (m1 + m2).
Now, we can solve for the mass of the caboose, m2. Rearranging the equation, we get:
v2^2 = (0.84 * m1 * v1^2) / (m1 + m2).
v2^2 * (m1 + m2) = 0.84 * m1 * v1^2.
v2^2 * m1 + v2^2 * m2 = 0.84 * m1 * v1^2.
v2^2 * m2 = 0.84 * m1 * v1^2 - v2^2 * m1.
m2 = (0.84 * m1 * v1^2 - v2^2 * m1) / v2^2.
We know the values of m1, v1, v2, and we can substitute them into the equation above to solve for m2, which is the mass of the caboose.