At time `t=0`, a small ball is projected from point A with a velocity of `60m//s` at `60^@` angle with horizontal. Neglect atmospheric resistance and determine the two times `t_1 and t_2` when the velocity of the ball makes an angle of `45^@` with horizontal x-axis.

Explanation

To solve the problem of determining the two times \( t_1 \) and \( t_2 \) when the velocity of the ball makes an angle of \( 45^\circ \) with the horizontal x-axis, we can follow these steps:

### Step 1: Determine the initial velocity components
The initial velocity \( u \) of the ball is given as \( 60 \, \text{m/s} \) at an angle of \( 60^\circ \) with the horizontal.

- The horizontal component of the initial velocity \( u_x \):
\[ u_x = u \cos(60^\circ) = 60 \cos(60^\circ) = 60 \times \frac{1}{2} = 30 \, \text{m/s} \]

- The vertical component of the initial velocity \( u_y \):
\[ u_y = u \sin(60^\circ) = 60 \sin(60^\circ) = 60 \times \frac{\sqrt{3}}{2} = 30\sqrt{3} \, \text{m/s} \]

### Step 2: Set up the equations for the velocity at \( 45^\circ \)
At the times \( t_1 \) and \( t_2 \) when the ball makes a \( 45^\circ \) angle with the horizontal, the vertical component of the velocity \( v_y \) will equal the horizontal component \( v_x \).

Since \( v_x \) remains constant:
\[ v_x = u_x = 30 \, \text{m/s} \]

Thus, at \( 45^\circ \):
\[ v_y = v_x = 30 \, \text{m/s} \]

### Step 3: Use the equations of motion to find \( t_1 \) and \( t_2 \)
The vertical component of the velocity \( v_y \) at time \( t \) is given by:
\[ v_y = u_y - g t \]
where \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity).

For the first case (upward motion):
\[ 30 = 30\sqrt{3} - 10 t_1 \]
Rearranging gives:
\[ 10 t_1 = 30\sqrt{3} - 30 \]
\[ t_1 = \frac{30(\sqrt{3} - 1)}{10} = 3(\sqrt{3} - 1) \approx 2.19 \, \text{s} \]

For the second case (downward motion):
\[ -30 = 30\sqrt{3} - 10 t_2 \]
Rearranging gives:
\[ 10 t_2 = 30\sqrt{3} + 30 \]
\[ t_2 = \frac{30(\sqrt{3} + 1)}{10} = 3(\sqrt{3} + 1) \approx 8.2 \, \text{s} \]

### Final Answer
The two times when the velocity of the ball makes an angle of \( 45^\circ \) with the horizontal are:
\[ t_1 \approx 2.19 \, \text{s} \quad \text{and} \quad t_2 \approx 8.2 \, \text{s} \]