Question

now jed and kadia tackle a homework problem:
an object of mass ( m_1 = 20 mathrm{~kg} ) and velocity ( vec{v}_1 = 6.5 mathrm{~m} / mathrm{s} ) crashes into another object of mass ( m_2 = 5 mathrm{~kg} ) and velocity ( vec{v}_2 = -16.5 mathrm{~m} / mathrm{s} ). the two particles stick together as a result of the collision. because no external forces are acting, the collision does not change the total momentum of the system of two particles, so the principle of conservation of linear momentum applies.
m_1 v_{1 i}+m_2 v_{2 i}=left(m_1+m_2
ight) v_f
if jed and kadia use the one - dimensional conservation of momentum equation to find the final velocity (in ( mathrm{m} / mathrm{s} )) of the two joined objects after the collision, what do they obtain? (indicate the direction with the sign of your answer.)

part 10 of 10 - analyze
although the forces that the two objects exert on each other cannot change their total momentum, they can change the total kinetic energy in an inelastic collision such as the one being considered. all of the mechanical energy in the problem is kinetic energy. change in kinetic energy is given by
delta k=\frac{1}{2}left(m_1+m_2
ight) v_f^2-left(\frac{1}{2} m_1 v_{1 i}^2+\frac{1}{2} m_2 v_{2 i}^2
ight).
what do jed and kadia find for the change in the total mechanical energy of the system? (enter your answer in j.)

Explanation:

Step1: Substitute the values into the formula

Given \(m_1 = 20\space kg\), \(v_{1i}=6.5\space m/s\), \(m_2 = 5\space kg\), \(v_{2i}=- 16.5\space m/s\), \(v_f = 1.9\space m/s\) into \(\Delta K=\frac{1}{2}(m_1 + m_2)v_f^2-(\frac{1}{2}m_1v_{1i}^2+\frac{1}{2}m_2v_{2i}^2)\)

First, calculate \(\frac{1}{2}(m_1 + m_2)v_f^2\):
\(\frac{1}{2}(20 + 5)\times(1.9)^2=\frac{1}{2}\times25\times3.61 = 45.125\)

Then, calculate \(\frac{1}{2}m_1v_{1i}^2+\frac{1}{2}m_2v_{2i}^2\):
\(\frac{1}{2}\times20\times(6.5)^2+\frac{1}{2}\times5\times(-16.5)^2\)
\(=10\times42.25 + 2.5\times272.25\)
\(=422.5+680.625=1103.125\)

Step2: Calculate \(\Delta K\)

\(\Delta K=45.125 - 1103.125=-1058\)

Answer:

\(- 1058\space J\)