When you throw a pebble straight up with initial speed V, it reaches a maximum height H with no air resistance. At what speed should you throw it up vertically so it will go twice as high?

and

When you drop a pebble from height H, it reaches the ground with speed V if there is no air resistance. From what height should you drop it so it will reach the ground with twice speed?

The answer of first question:

The speed should be √2 times the initial speed.

v°=√2*vi

v° stands for new speed

vi stands for initial speed

The answer of second question:

The height should be four times the previous height.

h°=4*h

h° stands for new height

h stands for previous height

Explanation:

According to conservation of energy:

The energy can't be created or destroyed but it can transform from one form to another form of energy.

When a pebble throw upward it's kinetic energy converted into potential energy. So, according to law of conservation of energy

Kinetic energy = Potential energy

(1/2) m*v² = m*g*h

m stands for mass of pebble

v stands for velocity

g stands for gravitational constant

h stands for height

(1/2) m*v² = m*g*h

Multiply by 2 and divide by m on both sides

v² = 2*g*h

Take square root on both sides

v = √ (2*g*h) ... Equation (A).

If we want twice the height (h°=2*h), then the new velocity will be v°

Again using law of conservation of energy

Kinetic energy = Potential energy

(1/2) m*v°² = m*g*h°

v° stands for new velocity

h° stands for new height

put h°=2*h in above equation

(1/2) m*v°² = m*g*2*h

Multiply by 2 and divide by m on both sides

v°² = 2*g*2*h

Take square root on both sides

v° = √ (2*g*2*h)

v° = √2*√ (2*g*h)

from equation (A) we know that v=√ (2*g*h)

So the equation become

v° = √2*v

The speed should be √2 times the initial speed.

Explanation for second answer

According to law of conservation of energy

Kinetic energy = Potential energy

(1/2) * m*v² = m*g*h

Divide by m on both sides

(1/2) * v² = g*h

Divide by g on both sides

v² / (2*g) = h ... Equation (B)

If we want twice the speed (v°=2*v), then the new height will be h°

According to law of conservation of energy

Kinetic energy = Potential energy

(1/2) m*v°² = m*g*h°

v° stands for new velocity

h° stands for new height

(1/2) m*v°² = m*g*h°

put v° = 2*v in above equation

(1/2) m * (2*v) ²=m*g*h°

(1/2) m * 4*v²=m*g*h°

Divide by m on both sides

(1/2) 4*v²=g*h°

Divide by g on both sides

4 (v²/2*g) = h°

From equation (B) we know that v² / (2*g)

So the equation become

h° = 4*h

The height should be four times the previous height.