A cart of mass 200kg with a pole on it is moving on a smooth road with a constant speed. From the top of the pole there is a light sphere of mass m suspended by a almost massless string length 2m. The cart crashes into cart of mass 100kg and sticks to it. The sphere can go in circles round the top because of this. What Is least loss in ke during crash. (m

To determine the least loss in kinetic energy during the crash of the two carts, we first need to analyze the system before and after the collision. The key here is to understand how momentum is conserved and how kinetic energy is affected during an inelastic collision.

Initial Setup

We have two carts: one with a mass of 200 kg moving at a constant speed and another stationary cart with a mass of 100 kg. The light sphere of mass m is suspended from a pole on the first cart. When the two carts collide and stick together, we need to calculate the kinetic energy before and after the collision.

Initial Kinetic Energy

The initial kinetic energy (KE_initial) of the system is solely due to the moving cart. The formula for kinetic energy is:

• KE = (1/2) * m * v²

Let’s denote the speed of the 200 kg cart as v. Therefore, the initial kinetic energy is:

• KE_initial = (1/2) * 200 kg * v²

Final Kinetic Energy After Collision

After the collision, the two carts stick together, and their combined mass becomes:

• m_total = 200 kg + 100 kg = 300 kg

Using the principle of conservation of momentum, we can find the final velocity (v_f) of the combined carts. The initial momentum of the system is:

• P_initial = 200 kg * v

After the collision, the momentum is:

• P_final = 300 kg * v_f

Setting the initial momentum equal to the final momentum gives us:

• 200 kg * v = 300 kg * v_f

From this, we can solve for v_f:

• v_f = (200/300) * v = (2/3) * v

Calculating Final Kinetic Energy

The final kinetic energy (KE_final) of the system after the collision is:

• KE_final = (1/2) * 300 kg * (v_f)²

Substituting v_f into the equation gives:

• KE_final = (1/2) * 300 kg * ((2/3) * v)²
• KE_final = (1/2) * 300 kg * (4/9) * v²
• KE_final = (600/9) * v² = (200/3) * v²

Loss in Kinetic Energy

The loss in kinetic energy during the crash can now be calculated by subtracting the final kinetic energy from the initial kinetic energy:

• Loss in KE = KE_initial - KE_final
• Loss in KE = (100) * v² - (200/3) * v²
• Loss in KE = (300/3) * v² - (200/3) * v²
• Loss in KE = (100/3) * v²

Conclusion

Thus, the least loss in kinetic energy during the crash is given by:

• Loss in KE = (100/3) * v²

This result shows how the kinetic energy is affected by the inelastic nature of the collision, where some energy is transformed into other forms, such as heat and sound, rather than being conserved as kinetic energy.