A body is thrown up from a tower. If the ratio of distance to displacement is n, the ratio of time for upward and down-
ward motion is

When a body is thrown upwards from a tower, it experiences two distinct phases: the upward motion until it reaches its highest point and the downward motion as it falls back to the ground. To analyze the relationship between distance, displacement, and time, we can break down the problem step by step.

Understanding Distance and Displacement

First, let's clarify the concepts of distance and displacement:

• Distance is the total path length traveled by the body, regardless of direction.
• Displacement is the straight-line distance from the initial position to the final position, along with the direction.

In this scenario, if a body is thrown upwards from a height \( h \) and reaches a maximum height \( H \) before falling back down, the total distance traveled during the entire motion is the sum of the upward distance and the downward distance. The displacement, however, is simply the difference between the initial height and the final height.

Establishing the Ratios

Let’s denote:

• Initial height of the tower = \( h \)
• Maximum height reached = \( H \)
• Total distance traveled = \( h + (H - h) = H \)
• Displacement = \( H - h \)

Given that the ratio of distance to displacement is \( n \), we can express this mathematically:

Distance/Displacement = n

Substituting the values, we have:

\( \frac{H}{H - h} = n \)

From this equation, we can derive the relationship between \( H \) and \( h \).

Analyzing Time Ratios

Next, we need to find the ratio of time taken for the upward motion (\( t_u \)) and the downward motion (\( t_d \)). The time taken to reach the maximum height can be determined using the kinematic equations. Assuming the initial velocity is \( u \) and the acceleration due to gravity is \( g \), the time to reach the highest point is given by:

\( t_u = \frac{u}{g} \)

For the downward motion, the time taken to fall from height \( H \) back to the ground can be calculated using the equation:

\( H = \frac{1}{2} g t_d^2 \)

Rearranging this gives:

\( t_d = \sqrt{\frac{2H}{g}} \)

Finding the Ratio of Times

Now, we can find the ratio of the time for upward motion to the time for downward motion:

Time Ratio = \( \frac{t_u}{t_d} = \frac{\frac{u}{g}}{\sqrt{\frac{2H}{g}}} = \frac{u}{\sqrt{2gH}} \)

To express this in terms of the ratio \( n \), we can substitute \( H \) from our earlier equation. However, without specific values for \( u \) and \( g \), we can conclude that the ratio of time for upward and downward motion depends on the initial velocity and the height from which the body is thrown.

Final Thoughts

In summary, the ratio of time for upward and downward motion is influenced by the initial velocity and the height of the tower. By understanding the relationship between distance, displacement, and time, we can analyze the motion of the body effectively. This approach not only helps in solving similar problems but also deepens our grasp of kinematics in physics.