B. In step 3
They incorrectly solved for x. It should have been x=-3 and x=5
Explanation:
Given:
Solving for
:

where:

Integrating to get
with initial conditions
:

Integrating to get x with initial conditions x(0) = 0:

When t=T:


Answer: 211.059 m
Explanation:
We have the following data:
The angle at which the ball leaves the bat
The initial velocity of the ball
The acceleration due gravity
We need to find how far (horizontally) the ball travels in the air: 
Firstly we need to know this velocity has two components:
<u>Horizontally:</u>
(1)
(2)
<u>Vertically:</u>
(3)
(4)
On the other hand, when we talk about parabolic movement (as in this situation) the ball reaches its maximum height just in the middle of this parabola, when
and the time
is half the time it takes the complete parabolic path.
So, if we use the following equation, we will find
:
(5)
Isolating
:
(6)
(7)
(8)
Now that we have the time it takes to the ball to travel half of is path, we can find the total time
it takes the complete parabolic path, which is twice
:
(9)
With this result in mind, we can finally calculate how far the ball travels in the air:
(10)
Substituting (2) and (9) in (10):
(11)
Finally:
Answer:
c > √(2ab)
Explanation:
In this exercise we are asked to find the condition for c in such a way that the results have been real
The given equation is
½ a t² - c t + b = 0
we can see that this is a quadratic equation whose solution is
t = [c ±√(c² - 4 (½ a) b)] / 2
for the results to be real, the square root must be real, so the radicand must be greater than zero
c² -2a b > 0
c > √(2ab)
Answer:
Explanation:
a ) Between r = 0 and r = r₁
Electric field will be zero . It is so because no charge lies in between r = 0 and r = r₁ .
b ) From r = r₁ to r = r₂
At distance r , charge contained in the sphere of radius r
volume charge density x 4/3 π r³
q = Q x r³ / R³
Applying Gauss's law
4πr² E = q / ε₀
4πr² E = Q x r³ / ε₀R³
E= Q x r / (4πε₀R³)
E ∝ r .
c )
Outside of r = r₂
charge contained in the sphere of radius r = Q
Applying Gauss's law
4πr² E = q / ε₀
4πr² E = Q / ε₀
E = Q / 4πε₀r²
E ∝ 1 / r² .