The velocity vectors of the two balls can be represented as follows:
Ball 1: Velocity = (5 m/s)i + (5 m/s)j
Ball 2: Velocity = (2√3 m/s)i + (2 m/s)j
Here, i and j represent the unit vectors along the x and y axes, respectively.
To find the angle between the two velocities, we can use the dot product formula:
V₁ · V₂ = |V₁| |V₂| cos θ
Where V₁ and V₂ are the magnitudes of the velocities, and θ is the angle between them.
Let's calculate the magnitudes:
|V₁| = √((5 m/s)² + (5 m/s)²) = √(25 + 25) = √50 = 5√2 m/s
|V₂| = √((2√3 m/s)² + (2 m/s)²) = √(12 + 4) = √16 = 4 m/s
Substituting these values into the dot product formula:
(5√2 m/s)(4 m/s) cos θ = (5 m/s)(4 m/s) cos θ
20√2 cos θ = 20 cos θ
Dividing both sides by 20 and simplifying:
√2 cos θ = cos θ
Since both sides are equal, we can conclude that cos θ = 1/√2 = √2/2.
Now, we need to find the angle θ such that cos θ = √2/2. This angle is 45 degrees.
Therefore, the angle between the motion of the two billiard balls is 45 degrees (option 4 is the closest with 15°, but the correct answer is 45°).