When you throw a ball, the work you do to accelerate it equals the kinetic energy the ball gains. If you do twice as much work when throwing the ball, does it go twice as fast? Explain. Yes. Twice as much work will give the ball twice as much kinetic energy. Since KE is proportional to the speed, the speed will double as well. Yes. Twice as much work will give the ball four times as much kinetic energy. Since KE is proportional to the speed squared, the speed will be the square root of 4, or twice as fast. No. Twice as much work will give the ball four times as much kinetic energy. Since KE is proportional to the speed, the speed will be four times larger. No. Twice as much work will give the ball twice as much kinetic energy. But since KE is proportional to the speed squared, the speed will be 2 times larger.

No. Twice as much work will give the ball twice as much kinetic energy. But since KE is proportional to the speed squared, the speed will be [tex]sqrt{2}[/tex] times larger.

Explanation:

The work done on the ball is equal to the kinetic energy gained by the ball:

[tex]W=K[/tex]

So when the work done doubles, the kinetic energy doubles as well:

[tex]2W = 2 K[/tex]

However, the kinetic energy is given by

[tex]K=\frac{1}{2}mv^2[/tex]

where

m is the mass of the ball

v is its speed

We see that the kinetic energy is proportional to the square of the speed, [tex]v^2[/tex]. We can rewrite the last equation as

[tex]v=\sqrt{\frac{2K}{m}}[/tex]

which also means

[tex]v=\sqrt{\frac{2W}{m}}[/tex]

If the work is doubled,

[tex]W'=2W[/tex]

So the new speed is

[tex]v'=\sqrt{\frac{2(2W)}{m}}=\sqrt{2}\sqrt{\frac{2W}{m}}=\sqrt{2} v[/tex]

So, the speed is [tex]\sqrt{2}[/tex] times larger.