To tackle this problem, we need to analyze the dynamics of the two blocks connected by a spring. The key here is to understand how the spring affects the motion of both blocks when they are in motion and how their velocities change due to the interaction through the spring. Let’s break down the information provided and see which statements are true.
Understanding the System
We have two blocks: one with a mass of 2 kg and the other with a mass of 5 kg. They are connected by a spring with a force constant of 100 N/m. Initially, the 2 kg block has a velocity of 4 m/s, and the 5 kg block has a velocity of -3 m/s when the spring is at its natural length. This means there is no compression or extension in the spring at that moment.
Analyzing the Statements
Now, let’s evaluate each statement one by one:
- (a) The velocity of the 2 kg block is maximum when the 5 kg block is at instantaneous rest.
This statement is true. When the spring is at its natural length, the blocks are not exerting any force on each other. As the spring compresses or stretches, it will exert forces that can change the velocities of both blocks. The 2 kg block will reach its maximum velocity when the 5 kg block is momentarily at rest, as all the kinetic energy will be transferred to the 2 kg block at that instant.
- (b) The maximum and minimum velocity of the 2 kg block is 7 m/s and -3 m/s respectively.
This statement is also true. The maximum velocity of the 2 kg block can be calculated using conservation of momentum and energy principles. When the spring is fully compressed or stretched, the velocities will change, but the maximum velocity can indeed reach 7 m/s under the right conditions, while the minimum velocity can drop to -3 m/s when the spring is fully extended in the opposite direction.
- (c) The maximum and minimum velocity of the 5 kg block is 4 m/s and zero respectively.
This statement is true as well. The maximum velocity of the 5 kg block can be 4 m/s when the spring is at its natural length, and it can momentarily come to rest (zero velocity) when the spring is fully compressed or stretched, depending on the energy transfer between the blocks.
- (d) All of the above.
Since all the previous statements are true, this statement is also correct. Each of the conditions described accurately reflects the behavior of the blocks connected by the spring.
Conclusion
In summary, all the statements (a), (b), (c), and (d) are correct. The dynamics of the system can be understood through the principles of conservation of momentum and energy, as well as the behavior of springs in motion. This problem illustrates the intricate relationship between mass, velocity, and spring force in a connected system.
