Introduction to Perfectly Inelastic Collisions
In physics, a collision where two objects stick together after impact is known as a perfectly inelastic collision. While momentum is always conserved in a closed system, kinetic energy is typically lost during these collisions, transforming into heat, sound, or deformation energy.
The Physics Problem
We are given:
- Mass of ball 1 ($m_1$) = 4 kg
- Velocity of ball 1 ($u_1$) = 10 m/s
- Mass of ball 2 ($m_2$) = 16 kg
- Velocity of ball 2 ($u_2$) = -4 m/s (negative because it is in the opposite direction)
Step 1: Conservation of Momentum
To find the final velocity ($v$) of the combined mass ($m_1 + m_2$), we use the principle of conservation of linear momentum:
$$m_1 u_1 + m_2 u_2 = (m_1 + m_2) v$$ $$(4 \times 10) + (16 \times -4) = (4 + 16) v$$ $$40 - 64 = 20 v$$ $$-24 = 20 v$$ $$v = -1.2 \text{ m/s}$$
Step 2: Calculate Initial Kinetic Energy ($KE_i$)
$$KE_i = \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2$$ $$KE_i = \frac{1}{2} (4)(10)^2 + \frac{1}{2} (16)(-4)^2$$ $$KE_i = 200 + 128 = 328 \text{ J}$$
Step 3: Calculate Final Kinetic Energy ($KE_f$)
$$KE_f = \frac{1}{2} (m_1 + m_2) v^2$$ $$KE_f = \frac{1}{2} (20) (-1.2)^2$$ $$KE_f = 10 \times 1.44 = 14.4 \text{ J}$$
Step 4: Calculate Energy Loss ($\Delta KE$)
$$\Delta KE = KE_i - KE_f$$ $$\Delta KE = 328 - 14.4 = 313.6 \text{ J}$$
The total energy lost during the impact is 313.6 Joules.