A boy standing on a long railroad car throws a ball straight upwards. The car is moving on the horizontal road with an acceleration of `1m/s^2` and the projectioon velocity into vertical direction is 9.8 m/s. How far behind the boy will the ball fall on the car?

Explanation

To solve the problem step by step, we will analyze the motion of the ball and the railroad car separately, and then find out how far behind the boy the ball will fall on the car.

### Step 1: Determine the Time of Flight of the Ball
The ball is thrown straight upwards with an initial velocity \( u_y = 9.8 \, \text{m/s} \). The time of flight \( T \) for a projectile thrown vertically can be calculated using the formula:

\[ T = \frac{2u_y}{g} \]

where \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)).

Substituting the values:

\[ T = \frac{2 \times 9.8}{9.8} = 2 \, \text{s} \]

### Step 2: Calculate the Distance Traveled by the Car
The car is moving with a constant acceleration \( a = 1 \, \text{m/s}^2 \). The distance \( d \) traveled by the car during the time \( T \) can be calculated using the equation of motion:

\[ d = ut + \frac{1}{2} a t^2 \]

Since the initial velocity \( u \) of the car is not given, we will assume it to be \( 0 \) for simplicity (the problem does not specify it). Thus, the distance becomes:

\[ d = 0 \cdot T + \frac{1}{2} \cdot 1 \cdot (2)^2 = \frac{1}{2} \cdot 1 \cdot 4 = 2 \, \text{m} \]

### Step 3: Calculate the Distance Traveled by the Ball
The ball is thrown vertically, so it does not have any horizontal motion relative to the boy. Therefore, the horizontal distance traveled by the ball during the time of flight is \( 0 \, \text{m} \).

### Step 4: Determine How Far Behind the Boy the Ball Falls
The distance behind the boy where the ball lands on the car can be found by subtracting the distance traveled by the ball from the distance traveled by the car:

\[ \text{Distance behind the boy} = \text{Distance traveled by the car} - \text{Distance traveled by the ball} \]

Substituting the values:

\[ \text{Distance behind the boy} = 2 \, \text{m} - 0 \, \text{m} = 2 \, \text{m} \]

### Final Answer
The ball will fall **2 meters behind the boy** on the car.

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