There are two transparent lifts A and B The velocity of A is 20m/sec in upwards direction, and retardation 8m/sec2 (square) The velocity of lift B is 10m/sec in downward direction, with retardation 2m/sec2 (square)A bolt from the top of lift A falls Height of lift A is 16 meter How much distance would have the bolt travelled in respect with the man standing in lift B?

To solve this problem, we need to analyze the motion of both lifts and the bolt. We’ll start by determining how long it takes for the bolt to fall from lift A and how far lift B moves during that time. Then, we can calculate the distance the bolt travels relative to the man in lift B.

Understanding the Motion of Lift A

Lift A is moving upwards with an initial velocity of 20 m/s and has a retardation (deceleration) of 8 m/s². We can use the following kinematic equation to find the time it takes for the bolt to fall:

• Final Velocity (v) = Initial Velocity (u) + Acceleration (a) × Time (t)

In this case, the bolt will fall when it is released, so we can consider its initial velocity to be the same as lift A's upward velocity (20 m/s). The acceleration acting on the bolt is due to gravity, which is approximately -9.81 m/s² (downward). However, since lift A is decelerating, we need to account for that as well. The effective acceleration of the bolt is:

• Effective Acceleration = -9.81 m/s² - (-8 m/s²) = -1.81 m/s²

Calculating Time to Fall

We can use another kinematic equation to find the time it takes for the bolt to fall 16 meters:

• Distance (s) = Initial Velocity (u) × Time (t) + 0.5 × Acceleration (a) × Time²

Substituting the known values:

• s = 16 m
• u = 20 m/s
• a = -1.81 m/s²

Now we can set up the equation:

16 = 20t + 0.5 × (-1.81) × t²

This simplifies to:

0.905t² - 20t + 16 = 0

Using the quadratic formula, t can be calculated as:

• t = [20 ± √(20² - 4 × 0.905 × 16)] / (2 × 0.905)

Calculating the discriminant:

20² - 4 × 0.905 × 16 = 400 - 57.76 = 342.24

Now, substituting back into the formula:

t = [20 ± √342.24] / 1.81

Calculating the square root and solving for t gives us the time it takes for the bolt to fall.

Analyzing Lift B's Motion

Lift B is moving downwards with an initial velocity of 10 m/s and has a retardation of 2 m/s². We need to find out how far lift B moves during the time t we calculated earlier.

Using the same kinematic equation:

• s = ut + 0.5at²

For lift B:

• u = 10 m/s
• a = -2 m/s²

Substituting into the equation gives us:

s = 10t + 0.5 × (-2) × t²

This simplifies to:

s = 10t - t²

Calculating the Relative Distance

The total distance the bolt travels relative to the man in lift B is the distance the bolt falls plus the distance lift B moves downwards during that time:

Distance traveled by the bolt = 16 m (downwards)

Distance traveled by lift B = 10t - t² (downwards)

Thus, the total distance the bolt travels relative to lift B is:

Distance = 16 + (10t - t²)

By substituting the value of t we calculated earlier, we can find the exact distance. This will give us the distance the bolt has traveled with respect to the man standing in lift B.

In summary, the key steps involve calculating the time it takes for the bolt to fall, determining how far lift B moves during that time, and then combining these distances to find the relative distance traveled by the bolt. Each of these calculations relies on understanding the principles of kinematics and the effects of acceleration and deceleration on moving objects.