A 2000 kg car moving north at 25 m/s collides head-on with a 5000 kg truck moving south at 35 m/s. The car and truck stick together after the collision. How much mechanical energy is lost in the collision (find KE;-KE)? 1.25 * 10 ^ 6 * J 5.57 * 10 ^ 6 * J 2.57 * 10 ^ 6 * J 3.56 * 10 ^ 6 * J
The Correct Answer and Explanation is :
To solve for the mechanical energy lost in the collision, we can first determine the initial and final kinetic energy of the system and then compute the energy lost as the difference between them.
Step 1: Calculate the Initial Kinetic Energy (KE_initial)
We need to calculate the kinetic energy of both the car and the truck before the collision.
The formula for kinetic energy is: KE=12mv2KE = \frac{1}{2} m v^2
Where:
- mm is the mass of the object,
- vv is the velocity of the object.
For the car: KEcar initial=12×2000 kg×(25 m/s)2=0.5×2000×625=625000 JKE_{\text{car initial}} = \frac{1}{2} \times 2000 \, \text{kg} \times (25 \, \text{m/s})^2 = 0.5 \times 2000 \times 625 = 625000 \, \text{J}
For the truck: KEtruck initial=12×5000 kg×(35 m/s)2=0.5×5000×1225=3062500 JKE_{\text{truck initial}} = \frac{1}{2} \times 5000 \, \text{kg} \times (35 \, \text{m/s})^2 = 0.5 \times 5000 \times 1225 = 3062500 \, \text{J}
Total initial kinetic energy: KEinitial=KEcar initial+KEtruck initial=625000 J+3062500 J=3687500 JKE_{\text{initial}} = KE_{\text{car initial}} + KE_{\text{truck initial}} = 625000 \, \text{J} + 3062500 \, \text{J} = 3687500 \, \text{J}
Step 2: Calculate the Final Kinetic Energy (KE_final)
After the collision, the car and truck stick together, so their combined mass is: mtotal=2000 kg+5000 kg=7000 kgm_{\text{total}} = 2000 \, \text{kg} + 5000 \, \text{kg} = 7000 \, \text{kg}
The velocity of the combined mass can be determined using the principle of conservation of momentum. The initial momentum of the system is: pinitial=mcar×vcar initial+mtruck×vtruck initialp_{\text{initial}} = m_{\text{car}} \times v_{\text{car initial}} + m_{\text{truck}} \times v_{\text{truck initial}} pinitial=(2000 kg×25 m/s)+(5000 kg×(−35 m/s))=50000−175000=−125000 kg⋅m/sp_{\text{initial}} = (2000 \, \text{kg} \times 25 \, \text{m/s}) + (5000 \, \text{kg} \times (-35 \, \text{m/s})) = 50000 – 175000 = -125000 \, \text{kg} \cdot \text{m/s}
Since momentum is conserved, the final momentum pfinalp_{\text{final}} is the same: pfinal=mtotal×vfinalp_{\text{final}} = m_{\text{total}} \times v_{\text{final}} vfinal=pinitialmtotal=−125000 kg⋅m/s7000 kg=−17.86 m/sv_{\text{final}} = \frac{p_{\text{initial}}}{m_{\text{total}}} = \frac{-125000 \, \text{kg} \cdot \text{m/s}}{7000 \, \text{kg}} = -17.86 \, \text{m/s}
Now, we calculate the final kinetic energy: KEfinal=12×7000 kg×(−17.86 m/s)2=0.5×7000×318.7=1115450 JKE_{\text{final}} = \frac{1}{2} \times 7000 \, \text{kg} \times (-17.86 \, \text{m/s})^2 = 0.5 \times 7000 \times 318.7 = 1115450 \, \text{J}
Step 3: Calculate the Mechanical Energy Lost
Finally, the mechanical energy lost is the difference between the initial and final kinetic energies: Energy lost=KEinitial−KEfinal=3687500 J−1115450 J=2572050 J\text{Energy lost} = KE_{\text{initial}} – KE_{\text{final}} = 3687500 \, \text{J} – 1115450 \, \text{J} = 2572050 \, \text{J}
So, the mechanical energy lost in the collision is approximately: 2.57×106 J2.57 \times 10^6 \, \text{J}
Thus, the correct answer is 2.57 × 10^6 J.
Explanation
In this inelastic collision, where the car and truck stick together after the impact, kinetic energy is not conserved. This is because some of the initial kinetic energy is transformed into other forms of energy, such as heat, sound, or deformation of the vehicles. The total momentum is conserved, which allows us to calculate the final velocity of the combined mass. The energy lost is the difference between the total initial kinetic energy and the total final kinetic energy.