A particle is projected upwards from bottom of a tower, and another is projected downwards from top of that tower. Can you give me the displacement formula [i.e in terms of s=ut+0.5at^2]

When dealing with the motion of two particles—one projected upwards from the bottom of a tower and the other downwards from the top—it's essential to understand how their displacements can be described using the equations of motion. The formula you mentioned, \( s = ut + \frac{1}{2}at^2 \), is a fundamental equation in kinematics that helps us analyze the motion of objects under uniform acceleration, such as gravity.

Understanding the Scenario

Let's break down the situation. We have two particles:

• Particle A: Projected upwards from the base of the tower with an initial velocity \( u_A \).
• Particle B: Projected downwards from the top of the tower with an initial velocity \( u_B \).

Both particles are influenced by gravity, which acts downwards with an acceleration \( g \) (approximately \( 9.81 \, \text{m/s}^2 \)). For the sake of simplicity, let's assume the upward direction is positive.

Displacement of Particle A

For Particle A, which is moving upwards, the displacement \( s_A \) can be expressed as:

s_A = u_A t - \frac{1}{2} g t^2

Here, \( u_A t \) represents the distance it travels upwards due to its initial velocity, while \( -\frac{1}{2} g t^2 \) accounts for the downward acceleration due to gravity, which reduces its upward displacement over time.

Displacement of Particle B

For Particle B, which is moving downwards, the displacement \( s_B \) can be described as:

s_B = h - u_B t - \frac{1}{2} g t^2

In this case, \( h \) is the height of the tower. The term \( -u_B t \) indicates the distance it travels downwards due to its initial velocity, and \( -\frac{1}{2} g t^2 \) again reflects the effect of gravity acting in the same direction as the motion.

Combining the Displacements

To find the relationship between the two particles, we can set their displacements equal to each other at the moment they meet:

s_A + s_B = h

Substituting the expressions for \( s_A \) and \( s_B \) into this equation gives:

u_A t - \frac{1}{2} g t^2 + (h - u_B t - \frac{1}{2} g t^2) = h

Simplifying this equation allows us to analyze the conditions under which the two particles meet. This approach provides a comprehensive understanding of their motion and the effects of gravity on their displacements.

Example Calculation

Suppose the height of the tower \( h \) is 100 meters, Particle A is projected upwards with an initial velocity of 20 m/s, and Particle B is projected downwards with an initial velocity of 10 m/s. To find the time \( t \) when they meet, you would substitute these values into the combined displacement equation and solve for \( t \).

This method not only illustrates the application of the displacement formula but also emphasizes the importance of understanding the motion of multiple objects under the influence of gravity. By analyzing their displacements, we can predict their interactions and the dynamics of their motion effectively.