Work Power And Energy

Step 1: Analyze the energy transfer Given:
- The efficiency of the machine is .
- The total energy used is 12 J.
- The mass of the block is 2 kg.
The total energy used is , but the machine only uses of that energy to lift the block, so the effective energy that goes into raising the block is:
Step 2: Use the energy equation to calculate velocity The potential energy at the height to which the block is raised is equal to the effective energy used by the machine. The potential energy is given by: where:
- (mass of the block),
- (acceleration due to gravity),
- is the height the block is lifted. Using the relation , we can solve for : Now, the block falls through the same height , and its potential energy is converted into kinetic energy when it reaches the ground. The kinetic energy at the ground is: Equating the potential energy and kinetic energy: Solving for : Thus, the velocity of the block when it reaches the ground is .

The velocity of the particle can be found using the work-energy theorem, which relates the change in velocity to the area under the acceleration versus position curve.
Given that the velocity at is , and from the graph, the area under the curve between and is the work done,
which contributes to the change in velocity.
From the graph, we can estimate the area, which leads to a velocity of at .
Thus, the velocity at is:

Step 1: Use the relation between power, force, and velocity
Power
Step 2: Rearranging and integrating
Step 3: Use
Step 4: Identify the exponent
So,