To solve the problem, we will use the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. Here are the steps to find the magnitude of the force applied to the ball:
### Step 1: Identify the given data
- Mass of the ball, \( m = 0.2 \, \text{kg} \)
- Distance moved by hand while applying the force, \( s_1 = 0.2 \, \text{m} \)
- Additional height reached by the ball after leaving the hand, \( h = 2 \, \text{m} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)
### Step 2: Calculate the total distance the ball travels after leaving the hand
The total distance the ball travels after the hand releases it is the distance moved by the hand plus the height it rises after being thrown:
\[
s_2 = s_1 + h = 0.2 \, \text{m} + 2 \, \text{m} = 2.2 \, \text{m}
\]
### Step 3: Apply the work-energy theorem
According to the work-energy theorem:
\[
W_{\text{total}} = \Delta KE
\]
Where \( W_{\text{total}} \) is the total work done on the ball, and \( \Delta KE \) is the change in kinetic energy.
Since the ball starts from rest and comes to rest at the maximum height, the change in kinetic energy is:
\[
\Delta KE = KE_{\text{final}} - KE_{\text{initial}} = 0 - 0 = 0
\]
### Step 4: Calculate the work done by the hand and by gravity
The work done by the hand \( W_{\text{hand}} \) and the work done against gravity \( W_{\text{gravity}} \) can be expressed as:
\[
W_{\text{hand}} = F \cdot s_1
\]
\[
W_{\text{gravity}} = -mg \cdot s_2
\]
### Step 5: Set up the equation
Since the total work done is zero, we have:
\[
W_{\text{hand}} + W_{\text{gravity}} = 0
\]
Substituting the expressions for work:
\[
F \cdot s_1 - mg \cdot s_2 = 0
\]
### Step 6: Solve for the force \( F \)
Rearranging the equation gives:
\[
F \cdot s_1 = mg \cdot s_2
\]
Now substituting the known values:
\[
F \cdot 0.2 = 0.2 \cdot 10 \cdot 2.2
\]
Calculating the right side:
\[
F \cdot 0.2 = 0.2 \cdot 10 \cdot 2.2 = 4.4
\]
Now, divide both sides by 0.2:
\[
F = \frac{4.4}{0.2} = 22 \, \text{N}
\]
### Final Answer
The magnitude of the force applied is \( F = 22 \, \text{N} \).
---
