To solve part (B), we need to calculate how far up the incline the student travels. The energy considerations will guide us through this process.

Step 1: Energy conservation and work-energy principle

The initial energy of the system is the kinetic energy of the student just after losing contact with the spring. This energy is used to do two things:

1. Work against gravity to lift the student up the incline.
2. Work against friction.

The total initial energy is the student's kinetic energy: $KE = \frac{1}{2} m v^2$ where:

• $m = 100 \, \mathrm{kg}$
• $v = 14 \, \mathrm{m/s}$

The energy is expended in two ways:

1. Gravitational potential energy gained: $PE = m g h$ where $h$ is the height the student reaches.

2. Work done against friction: $W_{\text{friction}} = f_k d = \mu_k N d$ where:

   • $\mu_k = 0.16$ (coefficient of kinetic friction),
   • $N = m g \cos \theta$ (normal force),
   • $d$ is the distance traveled along the incline.

Step 2: Relating height $h$ and incline distance $d$

From trigonometry on the incline: $h = d \sin \theta$

Step 3: Combine equations

The total work-energy equation is: $\frac{1}{2} m v^2 = m g h + \mu_k m g \cos \theta \, d$ Substitute $h = d \sin \theta$: $\frac{1}{2} m v^2 = m g (d \sin \theta) + \mu_k m g \cos \theta \, d$

Factor out $m g d$: $\frac{1}{2} v^2 = g d (\sin \theta + \mu_k \cos \theta)$

Solve for $d$: $d = \frac{\frac{1}{2} v^2}{g (\sin \theta + \mu_k \cos \theta)}$

Step 4: Substitute the values

• $v = 14 \, \mathrm{m/s}$,
• $g = 9.8 \, \mathrm{m/s^2}$,
• $\theta = 30^\circ$,
• $\mu_k = 0.16$.

Calculate $\sin 30^\circ = 0.5$ and $\cos 30^\circ \approx 0.866$: $d = \frac{\frac{1}{2} (14)^2}{9.8 (0.5 + 0.16 \times 0.866)}$

Step 5: Perform the calculation

1. Calculate $\frac{1}{2} (14)^2 = 98$,
2. Calculate $0.5 + 0.16 \times 0.866 = 0.5 + 0.13856 = 0.63856$,
3. Multiply $9.8 \times 0.63856 \approx 6.258$,
4. Divide $98 / 6.258 \approx 15.65 \, \mathrm{m}$.

Final Answer:

$\Delta s = 15.7 \, \mathrm{m}$

Let me know if you'd like further clarification or details! 😊

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Related Questions:

1. How does the spring constant affect the initial velocity?
2. What happens if the incline angle is increased?
3. How would doubling the mass impact the distance traveled up the incline?
4. How does friction influence the energy lost in this scenario?
5. Can this system be solved using force analysis instead of energy?

Tip: Always check your trigonometric values and units when solving incline problems!