two balls are rolling on a flat smooth table . one ball has velocity ✓3j and i while the other has components 2i and 2j if both start moving simultaneously from same point the angle between their paths is

To find the angle between the paths of the two balls, we can use the concept of vectors and the dot product. Each ball's velocity can be represented as a vector, and the angle between two vectors can be calculated using the dot product formula. Let's break this down step by step.

Understanding the Velocity Vectors

The velocities of the two balls are given as:

• Ball 1: \( \vec{v_1} = \sqrt{3} \hat{j} + \hat{i} \)
• Ball 2: \( \vec{v_2} = 2 \hat{i} + 2 \hat{j} \)

Expressing the Vectors

We can rewrite these vectors in component form:

• Ball 1: \( \vec{v_1} = (1, \sqrt{3}) \)
• Ball 2: \( \vec{v_2} = (2, 2) \)

Calculating the Dot Product

The dot product of two vectors \( \vec{a} = (a_1, a_2) \) and \( \vec{b} = (b_1, b_2) \) is given by:

\( \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 \)

For our vectors:

\( \vec{v_1} \cdot \vec{v_2} = (1)(2) + (\sqrt{3})(2) = 2 + 2\sqrt{3} \)

Finding the Magnitudes of the Vectors

The magnitude of a vector \( \vec{v} = (x, y) \) is calculated as:

\( |\vec{v}| = \sqrt{x^2 + y^2} \)

Calculating the magnitudes:

• Magnitude of \( \vec{v_1} \): \( |\vec{v_1}| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \)
• Magnitude of \( \vec{v_2} \): \( |\vec{v_2}| = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \)

Using the Dot Product to Find the Angle

The angle \( \theta \) between two vectors can be found using the formula:

\( \cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} \)

Substituting our values:

\( \cos(\theta) = \frac{2 + 2\sqrt{3}}{(2)(2\sqrt{2})} = \frac{2 + 2\sqrt{3}}{4\sqrt{2}} = \frac{1 + \sqrt{3}}{2\sqrt{2}} \)

Calculating the Angle

To find the angle \( \theta \), we take the inverse cosine:

\( \theta = \cos^{-1}\left(\frac{1 + \sqrt{3}}{2\sqrt{2}}\right) \)

This expression gives us the angle between the paths of the two balls. You can use a calculator to find the numerical value of this angle.

Final Thoughts

By following these steps, you can determine the angle between the paths of any two moving objects represented by vectors. This method is widely applicable in physics and engineering, where understanding the interaction of forces and motions is crucial.