A ball has an initial velocity of 22.0 m/s but is accelerating at a rate of -1.0 m/s squared. After it has traveled 25.0 meters what is its new velocity? The answer is 20.8 m/s squared. I don't no how my teacher got that answer. please help as soon as possible.

Thank you

To find the new velocity of the ball after it has traveled a certain distance, we can use the formula:

v = u + at

Where:
v is the final velocity (what we are trying to find)
u is the initial velocity (22.0 m/s)
a is the acceleration (-1.0 m/s^2)
t is the time it takes to travel the given distance (unknown)

In this case, we are given the distance, 25.0 meters. To find the time, we can use the following kinematic equation:

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

Where:
s is the distance traveled (25.0 meters)
u is the initial velocity (22.0 m/s)
a is the acceleration (-1.0 m/s^2)
t is the time it takes to travel the given distance (unknown)

Rearranging the equation, we have:

(1/2)at^2 + ut - s = 0

Now we can substitute the given values into this equation and solve for t. Using the quadratic formula:

t = (-u ± √(u^2 - 4(1/2)a(-s))) / (2(1/2)a)

Simplifying this equation, we get:

t = (-u ± √(u^2 + 2as)) / a

Once we have the value of t, we can substitute it back into the first equation to find the final velocity, v:

v = u + at

Now, let's calculate the values step by step:

1. Calculate the time, t:
t = (-u ± √(u^2 + 2as)) / a
= (-(22.0) ± √((22.0)^2 + 2(-1.0)(25.0))) / (-1.0)
= (-22.0 ± √(484.0 + (-50.0))) / (-1.0)
= (-22.0 ± √(434.0)) / (-1.0)
= (-22.0 ± 20.8) / (-1.0)

Considering both the positive and negative values:
t1 = (-22.0 + 20.8) / (-1.0) = 1.2 / (-1.0) = -1.2s
t2 = (-22.0 - 20.8) / (-1.0) = -42.8 / (-1.0) = 42.8s

We discard the negative value since time cannot be negative in this context. Therefore, t = 42.8s.

2. Calculate the final velocity, v:
v = u + at
= 22.0 + (-1.0)*(42.8)
= 22.0 - 42.8
= -20.8 m/s

Note that the final velocity is negative, indicating that the ball is moving in the opposite direction to its initial velocity.

Therefore, the correct final velocity of the ball after traveling 25.0 meters is -20.8 m/s, not 20.8 m/s as given in your question. Please double-check with your teacher or refer to any additional information given in the problem statement.