A particle is projected upwards as shown in in a diagram find its position at any time t.

To determine the position of a particle projected upwards at any time \( t \), we can use the principles of kinematics, particularly the equations of motion under constant acceleration. In this case, the only force acting on the particle after it is projected is gravity, which acts downwards with an acceleration of approximately \( 9.81 \, \text{m/s}^2 \). Let's break this down step by step.

Understanding the Variables

First, we need to define some key variables:

• Initial velocity (u): This is the speed at which the particle is projected upwards.
• Acceleration (a): For a particle moving upwards against gravity, this will be \(-g\) (where \( g \approx 9.81 \, \text{m/s}^2 \)). The negative sign indicates that gravity is acting in the opposite direction to the motion.
• Time (t): This is the time elapsed since the particle was projected.
• Position (s): This is the height of the particle above its starting point at time \( t \).

The Equation of Motion

We can use the second equation of motion, which relates position, initial velocity, time, and acceleration:

s = ut + \frac{1}{2} a t^2

Substituting the values we have:

s = ut - \frac{1}{2} g t^2

Breaking Down the Equation

In this equation:

• The term \( ut \) represents the distance the particle would travel if it continued moving at the initial velocity without any opposing force.
• The term \( -\frac{1}{2} g t^2 \) accounts for the downward acceleration due to gravity, which reduces the height of the particle over time.

Example Calculation

Let’s say a particle is projected upwards with an initial velocity of \( 20 \, \text{m/s} \). We want to find its position after \( 2 \, \text{s} \).

Using the equation:

s = 20(2) - \frac{1}{2}(9.81)(2^2)

Calculating each term:

• First term: \( 20 \times 2 = 40 \, \text{m} \)
• Second term: \( \frac{1}{2} \times 9.81 \times 4 = 19.62 \, \text{m} \)

Now, substituting back into the equation:

s = 40 - 19.62 = 20.38 \, \text{m}

Interpreting the Result

This means that after \( 2 \, \text{s} \), the particle is at a height of \( 20.38 \, \text{m} \) above its starting point. As time progresses, the height will continue to change until the particle reaches its maximum height and then starts to fall back down.

In summary, by applying the kinematic equations and understanding the effects of gravity, we can effectively determine the position of a particle projected upwards at any given time. This approach not only helps in solving problems in physics but also enhances our understanding of motion in a gravitational field.