To solve the problem step by step, we need to calculate the displacement of the ball with respect to the ground during its flight.
### Step 1: Understand the Initial Conditions
- The elevator is moving upwards with an acceleration of \( a = 5 \, \text{m/s}^2 \).
- The initial velocity of the elevator \( u_e = 10 \, \text{m/s} \).
- The height of the floor of the elevator from the ground \( h_e = 50 \, \text{m} \).
- The height from which the ball is shot above the floor of the elevator \( h_b = 2 \, \text{m} \).
- The initial speed of the ball with respect to the elevator \( u_b = 15 \, \text{m/s} \).
- The acceleration due to gravity \( g = 10 \, \text{m/s}^2 \).
### Step 2: Calculate the Initial Velocity of the Ball with Respect to the Ground
The initial velocity of the ball with respect to the ground \( u \) is the sum of the initial velocity of the elevator and the initial velocity of the ball with respect to the elevator:
\[
u = u_e + u_b = 10 \, \text{m/s} + 15 \, \text{m/s} = 25 \, \text{m/s}
\]
### Step 3: Determine the Effective Acceleration Acting on the Ball
Since the elevator is accelerating upwards, the effective acceleration acting on the ball when it is shot upwards will be:
\[
a = -g - a_e = -10 \, \text{m/s}^2 - 5 \, \text{m/s}^2 = -15 \, \text{m/s}^2
\]
(Note: The negative sign indicates that the acceleration is acting downwards).
### Step 4: Calculate the Time of Flight
To find the time of flight \( t \) until the ball strikes the floor of the elevator, we can use the kinematic equation:
\[
s = ut + \frac{1}{2} a t^2
\]
where \( s \) is the displacement of the ball with respect to the elevator. Since the ball is shot from a height of 2 m above the floor of the elevator and will fall back to that floor, we can set \( s = -2 \, \text{m} \) (the ball moves downwards).
Substituting the known values:
\[
-2 = 25t - \frac{1}{2} (15)t^2
\]
This simplifies to:
\[
-2 = 25t - 7.5t^2
\]
Rearranging gives us:
\[
7.5t^2 - 25t - 2 = 0
\]
### Step 5: Solve the Quadratic Equation
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 7.5, b = -25, c = -2 \):
\[
t = \frac{25 \pm \sqrt{(-25)^2 - 4 \cdot 7.5 \cdot (-2)}}{2 \cdot 7.5}
\]
Calculating the discriminant:
\[
625 + 60 = 685
\]
Thus,
\[
t = \frac{25 \pm \sqrt{685}}{15}
\]
Calculating the value gives approximately \( t \approx 2.13 \, \text{s} \) (taking the positive root).
### Step 6: Calculate the Displacement of the Ball with Respect to the Ground
Now, we can find the displacement of the ball with respect to the ground during its flight:
\[
s_g = ut + \frac{1}{2} (-g - a_e) t^2
\]
Substituting the values:
\[
s_g = 25 \cdot 2.13 + \frac{1}{2} (-15) (2.13)^2
\]
Calculating:
\[
s_g = 25 \cdot 2.13 - 7.5 \cdot (4.5369) \approx 53.25 - 34.02 = 19.23 \, \text{m}
\]
### Step 7: Total Displacement from the Ground
Finally, we need to add the height of the elevator floor from the ground:
\[
\text{Total Displacement} = h_e + s_g = 50 + 19.23 = 69.23 \, \text{m}
\]
### Final Answer
The displacement of the ball with respect to the ground during its flight would be approximately \( 69.23 \, \text{m} \).
