A 55.0-kg skateboarder starts out with a speed of 1.80 m/s. He does `+80.0J` of work on himself by pushing with his feet against the ground. In addition, friction does `-265J` of work on him. In both cases, the forces doing the work are non-conservative. The final speed of the skateboarder is 6.00 m/s. Caculate the change `[ Delta (P.E.) = (P.E.)_(f)-(P.E.)_(0)]` in the gravitational potential energy .

Options

A. `-1246J`
B. `-1086J`
C. `-821`J
D. `-716`J

Answer

To calculate the change in gravitational potential energy (\( \Delta P.E. \)) of the skateboarder, we can follow these steps:

### Step 1: Calculate the initial kinetic energy (\( KE_i \))
The initial kinetic energy can be calculated using the formula:
\[ KE_i = \frac{1}{2} m v_i^2 \]
Where:
- \( m = 55.0 \, \text{kg} \) (mass of the skateboarder)
- \( v_i = 1.80 \, \text{m/s} \) (initial speed)

Calculating:
\[ KE_i = \frac{1}{2} \times 55.0 \, \text{kg} \times (1.80 \, \text{m/s})^2 \]
\[ KE_i = \frac{1}{2} \times 55.0 \times 3.24 = 89.1 \, \text{J} \]

### Step 2: Calculate the final kinetic energy (\( KE_f \))
The final kinetic energy can be calculated using the same formula:
\[ KE_f = \frac{1}{2} m v_f^2 \]
Where:
- \( v_f = 6.00 \, \text{m/s} \) (final speed)

Calculating:
\[ KE_f = \frac{1}{2} \times 55.0 \, \text{kg} \times (6.00 \, \text{m/s})^2 \]
\[ KE_f = \frac{1}{2} \times 55.0 \times 36 = 990 \, \text{J} \]

### Step 3: Calculate the change in kinetic energy (\( \Delta KE \))
The change in kinetic energy is given by:
\[ \Delta KE = KE_f - KE_i \]
Calculating:
\[ \Delta KE = 990 \, \text{J} - 89.1 \, \text{J} = 900.9 \, \text{J} \]

### Step 4: Calculate the total work done (\( W_{total} \))
The total work done on the skateboarder is the sum of the work done by the skateboarder and the work done against friction:
\[ W_{total} = W_{self} + W_{friction} \]
Where:
- \( W_{self} = +80.0 \, \text{J} \)
- \( W_{friction} = -265.0 \, \text{J} \)

Calculating:
\[ W_{total} = 80.0 \, \text{J} - 265.0 \, \text{J} = -185.0 \, \text{J} \]

### Step 5: Use the work-energy principle to find \( \Delta P.E. \)
According to the work-energy principle:
\[ W_{total} = \Delta KE + \Delta P.E. \]
Rearranging gives us:
\[ \Delta P.E. = W_{total} - \Delta KE \]
Calculating:
\[ \Delta P.E. = -185.0 \, \text{J} - 900.9 \, \text{J} = -1085.9 \, \text{J} \]

### Conclusion
The change in gravitational potential energy is:
\[ \Delta P.E. \approx -1086 \, \text{J} \]