A 2.00-kg object traveling east at 20.0 m/s collides with a 3.00-kg object traveling west at 10.0 m/s.After the collision,the 2.00-kg object has a velocity 5.00 m/s to the west.How much kinetic energy was lost during the collision?
A) 0.000 J
B) 458 J
C) 516 J
D) 91.7 J
E) 175 J

Question

Quizplus · Accepted Answer

The kinetic energy before the collision can be calculated using the formula $KE = \frac{1}{2}mv^2$ for each object and then summing them up. For the 2.00-kg object, $KE_1 = \frac{1}{2}(2.00\, \text{kg})(20.0\, \text{m/s})^2 = 400\, \text{J}$. For the 3.00-kg object, $KE_2 = \frac{1}{2}(3.00\, \text{kg})(10.0\, \text{m/s})^2 = 150\, \text{J}$. The total initial kinetic energy is $400\, \text{J} + 150\, \text{J} = 550\, \text{J}$.After the collision, the 2.00-kg object has a velocity of 5.00 m/s to the west. The kinetic energy of this object is $KE_{1, \text{after}} = \frac{1}{2}(2.00\, \text{kg})(5.00\, \text{m/s})^2 = 25\, \text{J}$. To find the velocity of the 3.00-kg object after the collision, we use the conservation of momentum, since momentum is conserved in collisions. The initial momentum is $p_{\text{initial}} = (2.00\, \text{kg})(20.0\, \text{m/s}) + (3.00\, \text{kg})(-10.0\, \text{m/s}) = 40\, \text{kg}\cdot\text{m/s} - 30\, \text{kg}\cdot\text{m/s} = 10\, \text{kg}\cdot\text{m/s}$.After the collision, the momentum of the 2.00-kg object is $(2.00\, \text{kg})(-5.00\, \text{m/s}) = -10\, \text{kg}\cdot\text{m/s}$. To conserve momentum, the 3.00-kg object must have a momentum of $10\, \text{kg}\cdot\text{m/s} + 10\, \text{kg}\cdot\text{m/s} = 20\, \text{kg}\cdot\text{m/s}$, giving it a velocity of $20\, \text{kg}\cdot\text{m/s} / 3.00\, \text{kg} = 6.67\, \text{m/s}$. The kinetic energy of the 3.00-kg object after the collision is $KE_{2, \text{after}} = \frac{1}{2}(3.00\, \text{kg})(6.67\, \text{m/s})^2 \approx 66.7\, \text{J}$.The total kinetic energy after the collision is $25\, \text{J} + 66.7\, \text{J} = 91.7\, \text{J}$. The kinetic energy lost during the collision is $550\, \text{J} - 91.7\, \text{J} = 458.3\, \text{J}$, which rounds to $458\, \text{J}$, making option B correct.

A 200-Kg Object Traveling East at 20