If it takes total work W to give an object a speed v and kinetic energy K, starting from rest, what will be the object's speed (in terms of v) and kinetic energy (in terms of K) if... If it takes total work W to give an object a speed v and kinetic energy K, starting from rest, what will be the object's speed (in terms of v) and kinetic energy (in terms of K) if we do twice the work on it, again starting from rest? Read more

Steps to Solve

1. Understand Work and Kinetic Energy Relationship

The work done on an object is related to its kinetic energy (KE) by the equation: $$ W = \Delta KE $$ Where the initial kinetic energy is zero (since it starts from rest), this means: $$ W = KE = \frac{1}{2} mv^2 $$

2. Initial Work and Kinetic Energy

Let the total work done to give the object speed $v$ and kinetic energy $K$ be $W = K$. From the equation, we have: $$ K = \frac{1}{2} mv^2 $$

3. Doubling the Work

Now, when we double the work done, we have: $$ 2W = 2K $$

4. Setting Up the New Kinetic Energy Equation

When we apply double the work, we can set up the kinetic energy equation as follows: $$ 2K = \frac{1}{2} mv'^2 $$ Where $v'$ is the new speed.

5. Solving for the New Speed

Setting the two kinetic energy expressions equal gives: $$ 2K = \frac{1}{2} mv'^2 $$ Substituting $K = \frac{1}{2} mv^2$ yields: $$ 2 \left(\frac{1}{2} mv^2\right) = \frac{1}{2} mv'^2 $$

This simplifies to: $$ mv^2 = \frac{1}{2} mv'^2 $$

6. Isolate and Solve for New Speed

We can simplify this by eliminating $m$ from both sides: $$ v^2 = \frac{1}{2} v'^2 $$ Multiplying both sides by 2 gives: $$ 2v^2 = v'^2 $$

7. Calculate New Speed

Taking the square root of both sides provides: $$ v' = \sqrt{2} v $$

8. Determine New Kinetic Energy

For the new kinetic energy, substituting $v' = \sqrt{2} v$ into the kinetic energy formula gives: $$ KE' = \frac{1}{2} m (\sqrt{2} v)^2 = \frac{1}{2} m (2v^2) = 2\left(\frac{1}{2} mv^2\right) = 2K $$