What is the velocity at the midway point of a ball able to reach a height y when thrown with an initial velocity

v0?
(Assume the ball is thrown upward and that up is the positive direction. Use the following as necessary: y and g. )

Someone posted this question before and was given the answer

v= sqrt (v_o)^2 + 9.8y

This answer is apparently wrong.

Yes

To find the velocity at the midway point of a ball thrown upward with an initial velocity v0 and reaching a height y, we can make use of the principles of projectile motion and conservation of energy.

First, let's consider the motion of the ball at the midway point. At this point, the ball has traveled half the distance to its maximum height and is about to start descending.

To determine the velocity at this point, we can use the conservation of energy:

Potential energy at the maximum height = Kinetic energy at the midway point

The potential energy at the maximum height is given by:

Potential energy = m * g * y

Where m is the mass of the ball and g is the acceleration due to gravity (9.8 m/s^2).

The kinetic energy at the midway point is given by:

Kinetic energy = (1/2) * m * (velocity^2)

Now, equating the potential and kinetic energies, we get:

Potential energy = Kinetic energy
m * g * y = (1/2) * m * (velocity^2)

Simplifying the equation and solving for the velocity:

velocity^2 = 2 * g * y
velocity = sqrt(2 * g * y)

Therefore, the correct answer for the velocity at the midway point is:

velocity = sqrt(2 * g * y)

Maybe it is a reading of the question. If the max height is y, midpoint is y/2

If that is so, then the 9.8 should be -9.8, and the equation is dead on correct.

this came from a test bank so the second answer listed is correct but wasn't an option (you need the negative sign on g), so you had to do what the third person did so that h isn't in the answer. But, h was given in the original question so it should be allowed in the answer and I like the 2nd answer better.

The motion to the highest point (height “H”)

v = vₒ - g•t,
v=0, t = vₒ/g.
H = vₒ•t - g•t²/2 = vₒ• vₒ/g –(g/2) •(vₒ/g)² = vₒ²/2•g.
Velocity at the midpoint (height “h“)
v =vₒ - g•t1,
t1 =(vₒ- v)/g.
h = vₒ•t1 - g•t1²/2 =
=vₒ•(vₒ- v)/g - g•(vₒ- v)²/2•g² =
= [2•vₒ•(vₒ- v) -(vₒ- v)²]/ 2•g =
={2vₒ² - 2vₒ•v - vₒ² +2 vₒ•v - v²}/2•g =
= (vₒ² - v²)/2•g.
h=H/2 = vₒ²/2•2•g.
(vₒ² - v²)/2•g = vₒ²/2•2•g.
vₒ² = 2•v²,
v =vₒ/√2 = vₒ/1.41 =0.707•vₒ