**Projectile Motion Problem: Finding the Time at which the Velocity makes an Angle of 45 Degrees with the Horizontal**

To solve this problem, we need to use the equations of motion for projectile motion, which describe the motion of an object that is thrown or launched into the air at an angle. The equations of motion are based on the following assumptions:

- The object is a point particle, meaning it has no size or shape.
- The only force acting on the object is gravity, which is constant and acts in the downward direction.
- Air resistance is negligible.

**Step 1: Break the Initial Velocity into its x and y Components**

The first step in solving this problem is to break the initial velocity of the projectile into its x and y components. We can do this using trigonometry, since we know the angle at which the projectile is launched and the magnitude of its velocity.

vx = v * cos(θ)
vy = v * sin(θ)

where vx is the x-component of the velocity, vy is the y-component of the velocity, v is the magnitude of the velocity (50 m/s in this case), and θ is the launch angle (60 degrees in this case).

Substituting the values, we get:

vx = 50 m/s * cos(60°) = 25 m/s
vy = 50 m/s * sin(60°) = 43.3 m/s

**Step 2: Find the Time at which the Velocity makes an Angle of 45 Degrees with the Horizontal**

The next step is to find the time at which the velocity of the projectile makes an angle of 45 degrees with the horizontal. We can do this by using the equation for the angle between the velocity and the horizontal:

tan(θ') = vy / vx

where θ' is the angle between the velocity and the horizontal (45 degrees in this case).

Substituting the values, we get:

tan(45°) = 43.3 m/s / 25 m/s
1 = 1.732

This equation has no real solutions, which means that the velocity of the projectile never makes an angle of 45 degrees with the horizontal. Therefore, the answer to the problem is that there is no time at which this occurs.

**Step 3: Conclusion**

In conclusion, we have used the equations of motion for projectile motion to solve a problem involving the time at which the velocity of a projectile makes an angle of 45 degrees with the horizontal. We found that there is no such time, which means that the velocity of the projectile never makes this angle. This problem demonstrates the importance of breaking the initial velocity into its x and y components, which allows us to analyze the motion of the projectile in two dimensions.