A body starts slipping down an incline and moves half meter in half second. How long will it take to move the next half meter?

To determine how long it will take for the body to move the next half meter down the incline, we can analyze the motion using the principles of kinematics. Since the body is already in motion, we can assume it is accelerating due to gravity, which will affect the time it takes to cover the next distance.

Understanding the Motion

When an object slides down an incline, it accelerates due to the gravitational force acting on it. The distance it travels in a given time can be described using the equations of motion. In this case, we know that the body has moved 0.5 meters in 0.5 seconds. We can use this information to find the acceleration and then calculate the time for the next half meter.

Calculating the Initial Velocity

First, we need to find the average velocity during the first half meter. The average velocity (\(v_{avg}\)) can be calculated using the formula:

• Average Velocity: \(v_{avg} = \frac{d}{t}\)

Where \(d\) is the distance (0.5 meters) and \(t\) is the time (0.5 seconds).

Plugging in the values:

• \(v_{avg} = \frac{0.5 \text{ m}}{0.5 \text{ s}} = 1 \text{ m/s}\)

Finding the Acceleration

Next, we can assume that the body started from rest (initial velocity \(u = 0\)) and reached this average velocity. To find the acceleration (\(a\)), we can use the formula:

• Acceleration: \(a = \frac{v - u}{t}\)

Since the average velocity is 1 m/s and the initial velocity is 0 m/s, we can rearrange the formula to find \(a\):

• \(a = \frac{1 \text{ m/s} - 0 \text{ m/s}}{0.5 \text{ s}} = 2 \text{ m/s}^2\)

Calculating Time for the Next Half Meter

Now that we have the acceleration, we can find out how long it will take to travel the next half meter. We can use the second equation of motion:

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

For the next half meter, we know:

• Initial velocity (\(u\)) = 1 m/s (the velocity at the end of the first half meter)
• Distance (\(s\)) = 0.5 m

Substituting the known values into the equation:

• 0.5 = (1)t + \frac{1}{2}(2)t^2

This simplifies to:

• 0.5 = t + t^2

Rearranging gives us:

• t^2 + t - 0.5 = 0

Solving the Quadratic Equation

We can solve this quadratic equation using the quadratic formula:

• Quadratic Formula: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In our case, \(a = 1\), \(b = 1\), and \(c = -0.5\). Plugging in these values:

• t = \(\frac{-1 \pm \sqrt{1^2 - 4(1)(-0.5)}}{2(1)}\)

Calculating the discriminant:

• t = \(\frac{-1 \pm \sqrt{1 + 2}}{2}\)

Which simplifies to:

• t = \(\frac{-1 \pm \sqrt{3}}{2}\)

Since time cannot be negative, we take the positive root:

• t ≈ 0.366 seconds

Final Thoughts

Therefore, it will take approximately 0.366 seconds for the body to move the next half meter down the incline. This example illustrates how acceleration affects the time taken to cover distances in motion, especially when dealing with inclined planes and gravitational forces.