A 5 kg mass is raised distance of `4m` by a vertical force of 80 N. Find the final kinetic energy of the mass if it was originally at rest.`g =10 m//s^(2)`.

Explanation

To solve the problem step by step, we will follow the principles of work, energy, and the work-energy theorem.

### Step 1: Identify and Write Down Given Values
- Mass (m) = 5 kg
- Distance (s) = 4 m
- Force (F) = 80 N
- Acceleration due to gravity (g) = 10 m/s²

### Step 2: Understand the Initial Conditions
The mass is initially at rest, which means:
- Initial kinetic energy (KE_initial) = 0 J

### Step 3: Use the Work-Energy Theorem
According to the work-energy theorem:
\[ \text{Total Work Done (W)} = \Delta KE = KE_{final} - KE_{initial} \]
Since the initial kinetic energy is zero, we have:
\[ W = KE_{final} \]

### Step 4: Calculate Work Done Against Gravity
The work done against gravity (W_mg) can be calculated using the formula:
\[ W_{mg} = m \cdot g \cdot s \cdot \cos(180^\circ) \]
Here, cos(180°) = -1 (since the force of gravity acts downward while the displacement is upward).

Substituting the values:
\[ W_{mg} = 5 \, \text{kg} \cdot 10 \, \text{m/s}^2 \cdot 4 \, \text{m} \cdot (-1) \]
\[ W_{mg} = -200 \, \text{J} \]

### Step 5: Calculate Work Done by the Applied Force
The work done by the applied force (W_f) can be calculated using:
\[ W_f = F \cdot s \cdot \cos(0^\circ) \]
Here, cos(0°) = 1 (since the force is in the same direction as the displacement).

Substituting the values:
\[ W_f = 80 \, \text{N} \cdot 4 \, \text{m} \cdot 1 \]
\[ W_f = 320 \, \text{J} \]

### Step 6: Calculate Total Work Done
Now, we can find the total work done (W_t):
\[ W_t = W_{mg} + W_f \]
\[ W_t = -200 \, \text{J} + 320 \, \text{J} \]
\[ W_t = 120 \, \text{J} \]

### Step 7: Find the Final Kinetic Energy
From the work-energy theorem, we know:
\[ KE_{final} = W_t \]
Thus,
\[ KE_{final} = 120 \, \text{J} \]

### Final Answer
The final kinetic energy of the mass is **120 Joules**.

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