Questions: A snowboarder of total mass, m starts from rest at a height, h₀, on a frictionless slope. She slides down the slope to the bottom, glides along a flat horizontal section, and then up a smaller slope to a horizontal plateau at a final height of hₜ=3.0 m. Finally, she encounters a k=6000 N / m spring which compresses by a displacement of Δx by the time it has brought the snowboard to a complete stop. Assume that both friction and air resistance are negligible. Derive a single, simplified algebraic expression for the amount, Δx, that the spring will be compressed in bringing the snowboard to a stop. Your expression for Δx should be in terms of some or all of the following variables: m, g, k, h₀, hfinal.

A snowboarder of total mass, m starts from rest at a height, h₀, on a frictionless slope. She slides down the slope to the bottom, glides along a flat horizontal section, and then up a smaller slope to a horizontal plateau at a final height of hₜ=3.0 m.
Finally, she encounters a k=6000 N / m spring which compresses by a displacement of Δx by the time it has brought the snowboard to a complete stop. Assume that both friction and air resistance are negligible. Derive a single, simplified algebraic expression for the amount, Δx, that the spring will be compressed in bringing the snowboard to a stop. Your expression for Δx should be in terms of some or all of the following variables: m, g, k, h₀, hfinal.

Transcript text: A snowboarder of total mass, m starts from rest at a height, $\mathrm{h}_{0}$, on a frictionless slope. She slides down the slope to the bottom, glides along a flat horizontal section, and then up a smaller slope to a horizontal plateau at a final height of $h_{t}=3.0 \mathrm{~m}$. Finally, she encounters a $k=6000 \mathrm{~N} / \mathrm{m}$ spring which compresses by a displacement of $\Delta x$ by the time it has brought the snowboard to a complete stop. Assume that both friction and air resistance are negligible. Derive a single, simplified algebraic expression for the amount, $\Delta x$, that the spring will be compressed in bringing the snowboard to a stop. Your expression for $\Delta x$ should be in terms of some or all of the following variables: $\mathrm{m}, \mathrm{g}, \mathrm{k}, \mathrm{h}_{0}, \mathrm{~h}_{\text {final }}$.

Solution

Solution Steps

Step 1: Apply Conservation of Energy

Since friction and air resistance are negligible, the total mechanical energy of the snowboarder is conserved. The initial energy is purely gravitational potential energy at height _h0_, and the final energy is entirely stored as elastic potential energy in the compressed spring.

Step 2: Set up the Energy Equation

The initial gravitational potential energy is _mgh0_. The final elastic potential energy is (1/2) _k_(Δx)². Setting these equal gives:

_mgh0_ = (1/2) _k_(Δx)²

Step 3: Account for Final Height

The problem states the snowboarder ends at a final height _hfinal_. This means the initial gravitational potential energy is converted into both elastic potential energy and gravitational potential energy at _hfinal_. The equation becomes:

_mgh0_ = _mghfinal_ + (1/2) _k_(Δx)²

Step 4: Solve for Δx

Rearrange the equation to solve for Δx:

_mg_( _h0_ - _hfinal_) = (1/2) _k_(Δx)²

Δx = sqrt( (2_mg_( _h0_ - _hfinal_)) / k )