if air resistence is to be considered while annalysing motion of a particle thrown vertically upwards, then prove the relationship between time of ascent &time of descent is t_ascent descent.....

When analyzing the motion of a particle thrown vertically upwards while considering air resistance, we find that the relationship between the time of ascent and the time of descent is not as straightforward as it is in a vacuum. In a vacuum, the time taken to ascend is equal to the time taken to descend, but with air resistance, this relationship changes. Let's delve into the details to understand this phenomenon better.

The Basics of Vertical Motion

In the absence of air resistance, when a particle is thrown upwards, it decelerates due to gravity until it reaches its maximum height, where its velocity becomes zero. It then accelerates downwards back to the original height. The time taken to ascend (t_ascent) is equal to the time taken to descend (t_descent), leading to the simple relationship:

• t_ascent = t_descent

Introducing Air Resistance

Now, when we factor in air resistance, the situation changes significantly. Air resistance acts in the opposite direction to the motion of the particle, and its effect is more pronounced at higher velocities. This means that during ascent, the particle experiences a net force that is less than the gravitational force, causing it to decelerate more rapidly than it would in a vacuum. Conversely, during descent, the particle also experiences air resistance, but now it is moving downwards, which alters the dynamics of the forces acting on it.

Mathematical Representation

To analyze the motion mathematically, we can use the following equations:

• During ascent: F_net = -mg - kv
• During descent: F_net = mg - kv

Here, m is the mass of the particle, g is the acceleration due to gravity, k is a constant representing air resistance, and v is the velocity of the particle.

Time of Ascent vs. Time of Descent

Due to the presence of air resistance, the time of ascent (t_ascent) will be longer than the time of descent (t_descent). This can be understood intuitively:

• During ascent, the particle is fighting against both gravity and air resistance, which slows it down more than gravity alone would.
• During descent, while gravity is still acting, the air resistance also acts against the motion, but the particle has already lost some speed due to the ascent, leading to a shorter descent time.

Conclusion on Time Relationship

As a result, we can conclude that the relationship between the time of ascent and the time of descent when air resistance is considered can be expressed as:

• t_ascent > t_descent

This relationship highlights the impact of air resistance on vertical motion, demonstrating that the dynamics of motion are significantly altered when external forces are at play. Understanding these concepts is crucial for accurately predicting the behavior of objects in real-world scenarios.