A hill is 100 m long and makes an angle of \( 12^{\circ} \) with the horizontal. As a \( 50-\mathrm{kg} \) jogger runs up the hill, how much work does gravity do on the jogger? 1) 49000 J 2) 10000 J 3) -10000 J 4) zero 5) -49000 J

To find the work done by gravity on the jogger, we can use the formula for work, which is \( W = F \cdot d \cdot \cos(\theta) \). In this case, the force due to gravity acting downward is \( F = mg \) where \( m \) is the mass in kg (50 kg) and \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)). The distance \( d \) jogged up the hill is \( 100 \, m \). First, we find the force of gravity: \[ F = 50 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 490.5 \, \text{N} \] Next, the jogger is moving up the hill, which means the angle \( \theta \) between the direction of gravitational force and the displacement is \( 180^{\circ} - 12^{\circ} = 168^{\circ} \). Thus, \( \cos(168^{\circ}) = -\cos(12^{\circ}) \). Now we can calculate the work done by gravity: \[ W = 490.5 \, \text{N} \times 100 \, \text{m} \times \cos(168^{\circ}) = 490.5 \, \text{N} \times 100 \, \text{m} \times (-0.9781) \approx -4800 \, \text{J} \] Thus, the work done by gravity on the jogger is negative, indicating that gravity is doing work against the direction of motion, confirming the answer option is: 5) -49000 J It's worth noting that the negative sign here indicates the energy is spent to overcome gravity, but the amount calculated corresponds more closely to answer choice 3, which mentions -10000 J. So, considering the potential discrepancies in calculations based on rounding, option 3 may be chosen in settings with coarse estimations. However, given the clear choices listed, my final answer must be from listed choices reflecting the determination made above. Hence, you're most accurately looking at: 3) -10000 J