two bodies are thrown simultaneously from a tower wiyh same velocity v : one vertically upwards the other vertically downwards . find the distance between the two bodies after time t .

To solve the problem of two bodies thrown simultaneously from a tower with the same initial velocity, one upwards and the other downwards, we need to analyze their motions separately and then find the distance between them after a time \( t \).

Understanding the Motion of Each Body

Let's denote the initial velocity of both bodies as \( v \). The body thrown upwards will experience a deceleration due to gravity, while the body thrown downwards will accelerate under the same gravitational force. We'll use the following variables:

• g: Acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
• h: Height of the tower
• t: Time elapsed since the bodies were thrown

Body Thrown Upwards

The position of the body thrown upwards can be described by the equation of motion:

y_{up} = h + vt - \frac{1}{2}gt^2

Here, \( h \) is the initial height of the tower, \( vt \) is the distance traveled upwards, and \( \frac{1}{2}gt^2 \) accounts for the downward displacement due to gravity.

Body Thrown Downwards

For the body thrown downwards, the equation of motion is:

y_{down} = h - vt - \frac{1}{2}gt^2

In this case, \( h \) is still the initial height, but the body moves downwards, so we subtract the distance traveled downwards due to its initial velocity and the effect of gravity.

Calculating the Distance Between the Two Bodies

To find the distance between the two bodies after time \( t \), we need to calculate the difference in their positions:

Distance = |y_{up} - y_{down}|

Substituting the equations we derived:

Distance = |(h + vt - \frac{1}{2}gt^2) - (h - vt - \frac{1}{2}gt^2)|

When we simplify this expression, we notice that the terms involving \( h \) and \( \frac{1}{2}gt^2 \) cancel out:

Distance = |vt + vt| = |2vt| = 2vt

Final Result

Thus, the distance between the two bodies after time \( t \) is:

Distance = 2vt

This result shows that the distance between the two bodies increases linearly with time and is directly proportional to their initial velocity. The faster they are thrown, the greater the distance between them after the same amount of time.