To solve the problem step by step, we need to analyze the motion of the body projected at an angle of 60 degrees and use the concepts of kinetic energy (KE) and potential energy (PE).
### Step 1: Understand the given information
- The body is projected at an angle of \(60^\circ\) with the horizontal.
- The kinetic energy at the maximum height is given as \(10 \, \text{J}\).
### Step 2: Determine the total mechanical energy
At the maximum height, the kinetic energy is \(10 \, \text{J}\). The total mechanical energy (TE) of the projectile remains constant throughout its motion. Thus, we can express the total energy as:
\[
TE = KE + PE
\]
At the maximum height, the potential energy (PE) is at its maximum, and the kinetic energy (KE) is given as \(10 \, \text{J}\).
### Step 3: Calculate the total energy
Let \(H_{max}\) be the maximum height reached by the projectile. At the maximum height:
\[
TE = KE + PE_{max}
\]
Since potential energy at the point of projection is considered zero, we can write:
\[
TE = KE_{max} + PE_{max}
\]
At the maximum height, the potential energy is given by:
\[
PE_{max} = mgh_{max}
\]
Thus, we can express the total energy as:
\[
TE = 10 \, \text{J} + mgh_{max}
\]
Since we need to find the total energy, we need to relate it to the initial kinetic energy when the body was projected.
### Step 4: Find the initial kinetic energy
The initial kinetic energy when the body is projected can be calculated using the formula:
\[
KE_{initial} = \frac{1}{2} mv^2
\]
However, we can also express the total energy in terms of the kinetic energy at maximum height:
\[
TE = KE_{initial} = KE_{max} + PE_{max}
\]
Assuming \(PE_{max} = mgh_{max}\), we can rearrange this to find:
\[
TE = 10 \, \text{J} + mgh_{max}
\]
### Step 5: Establish the relationship between PE and KE
We need to find the height \(h\) at which the potential energy equals the kinetic energy. Thus, we set:
\[
PE = KE
\]
Let \(PE = KE = 20 \, \text{J}\) (since total energy is \(40 \, \text{J}\)). Therefore, we can write:
\[
mgh = 20 \, \text{J}
\]
### Step 6: Relate heights
From the previous calculations, we know:
\[
mgh_{max} = 30 \, \text{J}
\]
Now, we can set up the ratio:
\[
\frac{h_{max}}{h} = \frac{30 \, \text{J}}{20 \, \text{J}} = \frac{3}{2}
\]
This implies:
\[
h = \frac{2}{3} h_{max}
\]
### Conclusion
Thus, the height at which the potential energy and kinetic energy are equal is:
\[
\boxed{\frac{2}{3} h_{max}}
\]
