Final answer:
The speed of the pendulum at a height of 0.4m, while conserving mechanical energy, is calculated to be 1.4 m/s, considering its initial potential and kinetic energy from a push at a height of 1.2m.
Step-by-step explanation:
The question concerns the conservation of mechanical energy in the context of a pendulum that is given an initial push. According to the principle of conservation of energy, the total mechanical energy (potential plus kinetic) in a closed system is conserved if there is no energy added to or removed from the system (ignoring air resistance and friction). In this case, the pendulum has initial kinetic energy due to the push and potential energy due to its height. As it swings down, its potential energy decreases while its kinetic energy increases, keeping the total mechanical energy constant.
At the start, the pendulum has potential energy from its height of 1.2m and kinetic energy from its initial speed of 2m/s. At the lower height of 0.4m, the potential energy has decreased, meaning the kinetic energy must have increased if there is no energy loss. Using the formula for potential energy (PE = mgh) and kinetic energy (KE = 1/2 mv^2), and setting the total energy at both heights equal to each other since it's conserved, we can solve for the pendulum's speed at the height of 0.4m.
Let's calculate the speeds using the given heights and the initial speed:
- Initial total energy at 1.2m: PE + KE = mgh + 1/2 mv^2
- Total energy at 0.4m: Same as the initial total energy because energy is conserved.
When we solve for the new kinetic energy (and hence the speed) at 0.4m, we find that the speed of the pendulum at that height is 1.4 m/s, which corresponds to option D in the multiple-choice answers.