Question 9.53: In Anchorage, collisions of a vehicle with a moose are so co......
In Anchorage, collisions of a vehicle with a moose are so common that they are referred to with the abbreviation MVC. Suppose a 1000 kg car slides into a stationary 500 kg moose on a very slippery road, with the moose being thrown through the windshield (a common MVC result). (a) What percent of the original kinetic energy is lost in the collision to other forms of energy? A similar danger occurs in Saudi Arabia because of camel-vehicle collisions (CVC). (b) What percent of the original kinetic energy is lost if the car hits a 300 kg camel? (c) Generally, does the percent loss increase or decrease if the animal mass decreases?
With an initial speed of \nu_{i}, the initial kinetic energy of the car is K_{i}=m_{e}\nu_{i}^{2}/2\,. After a totally inelastic collision with a moose of mass m_{m} , by momentum conservation, the speed of the combined system is
m_{c}\nu_{i}=(m_{c}+m_{m})\nu_{f}\ \ \Rightarrow\ \nu_{f}=\frac{m_{c}\nu_{i}}{m_{c}+m_{m}},
with final kinetic energy
K_{f}=\frac{1}{2}(m_{c}+m_{m})\nu_{f}^{2}=\frac{1}{2}{(m_{c}+m_{m})\bigg(\frac{m_{c}\nu_{i}}{m_{c}+m_{m}}\bigg)}^{2}=\frac{1}{2}\frac{m_{c}^{2}}{m_{c}+m_{m}}\nu_{i}^{2}.
(a) The percentage loss of kinetic energy due to collision is
\frac{\Delta K}{K_{i}}=\frac{K_{i}-K_{f}}{K_{i}}=1-\frac{K_{f}}{K_{i}}=1-\frac{m_{c}}{m_{c}+m_{m}}=\frac{m_m}{m_c+m_m}=\frac{500 kg}{1000\;\mathrm{kg}+500\;\mathrm{kg}}=\frac{1}{3}=33.3\ \%.
(b) If the collision were with a camel of mass m_{{camel}} = 300 kg, then the percentage loss of kinetic energy would be
{\frac{\Delta K}{K_{i}}}={\frac{m_{\mathrm{camel}}}{m_{c}+m_{\mathrm{camel}}}}={\frac{300\;{\mathrm{kg}}}{1000\;{\mathrm{kg}}+300\;{\mathrm{kg}}}}={\frac{3}{13}}=23\ \%.
(c) As the animal mass decreases, the percentage loss of kinetic energy also decreases.