A marble rolling on smooth floor has a initial velocity of 0.4 m/s if floor offers a retardation of 0.02 m/s2 calculate the time it will take to rest

To calculate the time it will take for the marble to come to rest, we can use the formula for motion with uniform retardation:

v^2 = u^2 - 2as

Where:
v = final velocity (0 m/s, since the marble comes to rest)
u = initial velocity (0.4 m/s)
a = retardation (-0.02 m/s^2, since it acts in the opposite direction of motion)
s = distance traveled (unknown)

Rearranging the formula, we get:

s = (u^2 - v^2) / (2a)

Plugging in the values, we have:

s = (0.4^2 - 0^2) / (2 * (-0.02))
s = 0.16 / (-0.04)
s = -4 meters

The negative sign indicates that the marble moves in the opposite direction of the initial velocity.

Now, we can use another formula to calculate the time it takes for the marble to travel this distance:

s = ut + (1/2)at^2

Since we're interested in the time it will take to rest, we set the final distance (s) to -4 meters, the initial velocity (u) to 0.4 m/s, and the acceleration (a) to -0.02 m/s^2.

-4 = 0.4t + (1/2)(-0.02)t^2

Rearranging and simplifying the equation, we have:

-0.02t^2 + 0.4t - 4 = 0

This is a quadratic equation that we can solve to find the time it will take for the marble to rest. Using the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

Where:
a = -0.02
b = 0.4
c = -4

Plugging in these values:

t = (-0.4 ± √(0.4^2 - 4 * (-0.02) * (-4))) / (2 * (-0.02))

Calculating further:

t = (-0.4 ± √(0.16 - 0.32)) / (-0.04)
t = (-0.4 ± √(-0.16)) / (-0.04)

Since the square root of a negative number is not real, there are no real solutions to the equation. In other words, the marble will never come to rest according to these given conditions.

Please note that these calculations assume that the retardation remains constant. In practice, other factors like friction may affect the marble's motion as well.

calculate velocity leaving the floor, Vi

then
v = Vi - g t
0 = Vi - 9.81 t

Now