To solve the problem step by step, let's break down the information given and the calculations needed.
### Step 1: Understand the motion of the ball
The boy throws the ball vertically upward with a velocity of \(10 \, \text{m/s}\) relative to himself. Since the truck is moving horizontally with a velocity of \(10 \, \text{m/s}\), the ball will also have a horizontal component of velocity equal to that of the truck.
### Step 2: Determine the horizontal and vertical components of the ball's motion
- **Horizontal velocity of the ball (\(v_{2}\))**: This is equal to the truck's velocity, which is \(10 \, \text{m/s}\).
- **Vertical velocity of the ball (\(v_{1}\))**: This is the velocity with which the boy throws the ball, which is \(10 \, \text{m/s}\) upward.
### Step 3: Calculate the time of flight of the ball
To find the time of flight, we first determine the time taken to reach the maximum height. At maximum height, the final vertical velocity (\(v_{1}'\)) becomes \(0 \, \text{m/s}\).
Using the equation of motion:
\[
v_{1}' = v_{1} + a \cdot t
\]
Where:
- \(v_{1}' = 0 \, \text{m/s}\) (final vertical velocity at max height)
- \(v_{1} = 10 \, \text{m/s}\) (initial vertical velocity)
- \(a = -g = -10 \, \text{m/s}^2\) (acceleration due to gravity, acting downwards)
Substituting the values:
\[
0 = 10 - 10 \cdot t
\]
Rearranging gives:
\[
10 \cdot t = 10 \implies t = 1 \, \text{s}
\]
This is the time to reach maximum height. The total time of flight (\(t_f\)) is double this time:
\[
t_f = 2 \cdot t = 2 \, \text{s}
\]
### Step 4: Calculate the horizontal distance traveled by the ball
The horizontal distance (\(\Delta S\)) traveled by the ball during the time of flight can be calculated using:
\[
\Delta S = v_{2} \cdot t_f
\]
Where:
- \(v_{2} = 10 \, \text{m/s}\) (horizontal velocity of the ball)
- \(t_f = 2 \, \text{s}\) (total time of flight)
Substituting the values:
\[
\Delta S = 10 \, \text{m/s} \cdot 2 \, \text{s} = 20 \, \text{m}
\]
### Step 5: Conclusion
The distance of the ball from the pole where it lands is \(20 \, \text{m}\).
### Summary of Steps:
1. Identify horizontal and vertical components of the ball's motion.
2. Calculate the time of flight using kinematic equations.
3. Calculate the horizontal distance traveled during the time of flight.
