A particle starts with initial speed u and retardation a to come to rest in time t . The time taken to cover first half of the total path travelled is

To find the time taken to cover the first half of the total distance traveled by a particle that starts with an initial speed \( u \) and comes to rest with a retardation \( a \) in time \( t \), we can break down the problem step by step.

Understanding the Motion

The particle is decelerating uniformly, meaning its speed decreases at a constant rate until it reaches zero. The total distance \( s \) covered during this time can be calculated using the formula for uniformly accelerated motion:

Distance Formula: \( s = ut + \frac{1}{2}(-a)t^2 \)

Since the particle comes to rest, we can express the total distance as:

Distance Traveled: \( s = ut - \frac{1}{2}at^2 \)

Calculating Total Distance

Now, substituting the values, we can find the total distance \( s \) that the particle travels before coming to rest:

Total Distance: \( s = ut - \frac{1}{2}at^2 \)

Finding Half of the Distance

The first half of the total distance is simply:

Half Distance: \( \frac{s}{2} = \frac{1}{2} \left( ut - \frac{1}{2}at^2 \right) \)

Using the Equations of Motion

To find the time taken to cover this half distance, we can use the equation of motion again. Let \( t_1 \) be the time taken to cover half the distance. The distance covered in time \( t_1 \) can be expressed as:

Distance in Time \( t_1 \): \( s_1 = ut_1 - \frac{1}{2}at_1^2 \)

Setting Up the Equation

We know that \( s_1 = \frac{s}{2} \), so we can set up the equation:

Substituting for \( s \):

\( ut_1 - \frac{1}{2}at_1^2 = \frac{1}{2} \left( ut - \frac{1}{2}at^2 \right) \)

Solving for \( t_1 \)

Now, we can rearrange this equation to solve for \( t_1 \). First, multiply through by 2 to eliminate the fraction:

\( 2ut_1 - at_1^2 = ut - \frac{1}{2}at^2 \)

Rearranging gives us a quadratic equation:

\( at_1^2 - 2ut_1 + \left( ut - \frac{1}{2}at^2 \right) = 0 \)

Using the quadratic formula \( t_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where:

• \( a = a \)
• \( b = -2u \)
• \( c = ut - \frac{1}{2}at^2 \)

Substituting these values into the quadratic formula will yield the time \( t_1 \) taken to cover the first half of the distance. You will find that there are two possible solutions, and you will choose the one that is physically meaningful (i.e., positive time).

Final Thoughts

This approach allows you to systematically determine the time taken to cover the first half of the distance traveled by the particle. By applying the equations of motion and understanding the relationship between distance, speed, and time, you can solve similar problems involving uniformly accelerated motion.