You run a race with your friend. At first you each have the same kinetic energy, but then you find that she is beating you. When you increase your speed by 21.0%, you are running at the same speed she is. If your mass is 77.0 kg, what is her mass?

To solve this problem, we can use the concept of conservation of kinetic energy.

Step 1: Understand the problem and identify the given information.
- You and your friend initially have the same kinetic energy.
- Your friend is beating you, indicating that she has more kinetic energy than you.
- When you increase your speed by 21.0%, you have the same kinetic energy as your friend.
- Your mass is given as 77.0 kg.

Step 2: Set up an equation using the conservation of kinetic energy.
The kinetic energy of an object is given by the equation:
KE = (1/2) * mass * velocity^2

Since the initial kinetic energy is the same for both you and your friend, we can write the following equation:
(1/2) * mass * velocity_initial^2 = (1/2) * mass_friend * velocity_friend^2

Step 3: Use the given information to solve for the mass of your friend.
Let's assume your initial speed is V_initial and your friend's initial speed is V_friend.

After increasing your speed by 21.0%, your new speed is:
V_new = V_initial + 0.21 * V_initial = 1.21 * V_initial

We know that at this new speed, you have the same kinetic energy as your friend, so we can rewrite the equation as follows:
(1/2) * mass * (1.21 * V_initial)^2 = (1/2) * mass_friend * V_friend^2

Step 4: Simplify the equation and solve for the mass of your friend.
Rearrange the equation and isolate the mass_friend:
(1.21)^2 * mass = mass_friend

Now, substitute the given value for your mass (77.0 kg) into the equation:
(1.21)^2 * 77.0 kg = mass_friend

Calculating this equation, we find:
mass_friend ≈ 113.5906 kg

Therefore, your friend's mass is approximately 113.5906 kg.