Calculate Train Acceleration to Catch Thrown Ball

To find the acceleration of the train, we analyze the motion of the ball in the non-inertial frame of reference of the train.

1. Motion relative to the Train

Let the train be accelerating in the horizontal $x$-direction with a constant acceleration $a$. In the frame of reference of the train, the pseudo-acceleration acting on the ball is $-a\hat{i}$, and the acceleration due to gravity is $-g\hat{j}$ (where $g = 10\text{ m/s}^2$).

• Initial velocity of the ball relative to the train:
  $u_{x, \text{rel}} = u \cos 60^\circ = 10 \times \frac{1}{2} = 5\text{ m/s}$
  $u_{y, \text{rel}} = u \sin 60^\circ = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3}\text{ m/s}$

• Accelerations relative to the train:
  $a_{x, \text{rel}} = -a$
  $a_{y, \text{rel}} = -g = -10\text{ m/s}^2$

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2. Time of Flight

The ball returns to its initial vertical height when its vertical displacement relative to the train becomes zero:
$y_{\text{rel}} = u_{y, \text{rel}} T - \frac{1}{2} g T^2 = 0$

Solving for the time of flight $T$ (for $T \neq 0$):
$T = \frac{2 u_{y, \text{rel}}}{g} = \frac{2 \times 5\sqrt{3}}{10} = \sqrt{3}\text{ s}$

Thus, $T^2 = 3\text{ s}^2$.

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3. Horizontal Displacement inside the Train

The boy has to move forward by $1.15\text{ m}$ inside the train to catch the ball. Therefore, the horizontal displacement of the ball relative to the train during the time of flight $T$ is $x_{\text{rel}} = 1.15\text{ m}$.

Using the kinematic equation for horizontal motion relative to the train:
$x_{\text{rel}} = u_{x, \text{rel}} T + \frac{1}{2} a_{x, \text{rel}} T^2$

Substitute the known values into the equation:
$1.15 = 5 T - \frac{1}{2} a T^2$

Substitute $T = \sqrt{3}\text{ s}$ and $T^2 = 3$:
$1.15 = 5\sqrt{3} - \frac{3}{2} a$

Taking $\sqrt{3} \approx 1.732$:
$1.15 \approx 5(1.732) - 1.5 a$
$1.15 \approx 8.66 - 1.5 a$

Rearranging to solve for $a$:
$1.5 a = 8.66 - 1.15$
$1.5 a = 7.51 \approx 7.5$
$a = \frac{7.5}{1.5} = 5\text{ m/s}^2$

Thus, the acceleration of the train is $5\text{ m/s}^2$.