A circuit consists of a wire, a lightbulb, a switch, and a battery. What would happen if the battery were replaced with a section of wire?

No voltage, no current. The battery provides potential difference(v) to the circuit. In case of no potential difference, theres no current.

Given,
v is the first velocity
h is the first height obtained.

According to equation of motion is a straight line:

v = u - 2gh (- since the ball is thrown against gravity)

v= -2gh — (i) (u is 0 since the ball is thrown from rest)

Now,
Let V be new velocity
Let H be new height obtained

According to the question:
V = 2v

Again from the equation of motion in straight line,

V = u - 2gH
2v = 0 - 2gH (ball is thrown from rest so u is 0)

From eq — (i):

2(-2gh) = -2gH
-4gh = -2gH

Therefore H= 2h ( Since, -2g & -4g cancel each other)

Hence the new height is twice the old height.