The Hartwell Community Center hosts events on its second floor, and the fire marshal has flagged a problem: the only way out is two narrow stairwells. The center wants an emergency descent slide off its east balcony, 3.0 m above the ground. The marshal’s rule is that riders of any size, from a 40 kg student to a 100 kg adult, must leave the slide at no more than 4.0 m/s (about 9 mph) so nobody is hurt, and at no less than 1.5 m/s so riders clear the exit and the next person doesn’t pile up behind them. The center has $300. You decide the slide’s length, its surfaces, and whether to add a flat runout. Your job is to design something that provably works.
📊Reference Data & Standards
These are the prices and measured friction ratios for what the center can order. The friction ratio μ is the friction force divided by the rider’s weight, the same F_f ÷ weight you can measure with a spring scale, as you did in Activity 4-2-2.
Material / item
Cost
Friction ratio μ
What to know
Smooth vinyl sheeting
$18 per m
0.15
Very slippery. Fast.
Nylon slide cloth
$14 per m
0.25
A typical slide fabric.
Canvas tarp
$10 per m
0.35
Cheap, with moderate drag.
Felt-backed runner
$22 per m
0.50
Grippy. Can stall a rider on a gentle slope.
Rubberized mat
$28 per m
0.70
Very grippy. Best kept for flat runouts.
Frame and rails (under every section)
$15 per m
—
Added to the cost of every meter of slope and runout.
Landing pad at the exit
$40 flat
—
Required on every design.
Here is the energy budget for any slide design. Use it to check a design before you build or test anything.
x = √(s² − H²) is the horizontal distance of the slope, and tanθ = H ÷ x is its steepness. E_friction on the slope = μ_slope × m × g × x E_friction on the runout = μ_runout × m × g × r v = √( 2g × (H − μ_slope·x − μ_runout·r) ) Keep-sliding check: μ_slope < tanθ
Where does the x come from? On the slope, the friction force is F_f = μ·m·g·cosθ, and it acts over the slope length s. So the thermal energy is F_f × s = μ·m·g·(s cosθ), and s cosθ is exactly the horizontal distance x. On the flat runout, friction is simply μ·m·g over the runout length r.
Worked example: one possible design
Given
Balcony H = 3.0 m
Slope: s = 6.0 m of nylon slide cloth (μ = 0.25)
Runout: r = 2.0 m of felt-backed runner (μ = 0.50)
✓ This design works. A 40 kg rider starts with 1,176 J and a 100 kg rider with 2,940 J, but both exit at about 3.7 m/s.
📏Constraints
The drop is fixed: the balcony is H = 3.0 m above the ground.
Every rider from 40 kg to 100 kg must exit between 1.5 m/s and 4.0 m/s.
Total budget: $300. That includes the $40 landing pad and the $15-per-meter frame under every section.
The slope must be between 5.3 m and 9.0 m long (no steeper than 35°). A flat runout of 0 to 4 m is optional.
The rider must keep sliding on the slope: the slope surface’s friction ratio μ has to be less than tanθ (the slope’s steepness).
Friction ratios and costs must come from the Reference Data table, unless you measure and document your own.
🚚Deliverables
A final design: slope length, slope surface, runout length, and runout surface, with an itemized cost that fits the budget.
An energy budget showing where a 40 kg rider’s and a 100 kg rider’s starting energy goes, with an exit speed in the allowed range.
Evidence that checks your friction values: drag tests (if you model the design) or a 1:10 scale model (if you build it), with a percent error.
A written design memo explaining your choices and the tradeoffs you weighed.
🗓️Suggested Pacing
Day 1: Explore. Read the data, work the example by hand, and try designs in the Slide Designer. Choose one design to pursue.
Day 2: Test. Check your friction values with drag tests or a 1:10 scale model, and calculate your percent error.
Day 3: Finalize. Update the design with your measured values, confirm every constraint, and write your design memo.
🛝Build Your Design
Use this to explore designs before you write anything up. Work your chosen design by hand too. The Designer is a check, not a substitute for showing your calculations.
🛝 Slide Designer
Try designs here before you commit to one. Every check below is a constraint from the problem.
x = √(6² − 3²) = 5.20 m · tanθ = 0.58
✓ Slope angle is 30.0° (limit 35°)
✓ Rider keeps sliding: μ = 0.25 vs. tanθ = 0.58
✓ Rider reaches the exit with energy left
✗ Exit speed is 5.77 m/s (must be ≤ 4)
✓ Exit speed is at least 1.5 m/s
✓ Cost is $214 (budget $300)
Not there yet. Adjust the design until every check turns green.
Slide the rider mass from 40 to 100 kg. The energy numbers change, but the exit speed doesn’t.
🔬Test Your Friction Numbers
Your whole design rests on the friction ratios. Check them with one of these two options.
Modeled: drag tests. For each surface in your design, tape a sample to a board and drag a block across it with a spring scale at a steady speed. Hang the block from the scale to get its weight. Then μ = F_f ÷ weight. Compare it with the reference value and calculate the percent difference.
Built: a 1:10 scale model. Make a model slope with a 0.30 m rise and a length of s ÷ 10, covered with your slope material. Release a wooden block from rest at the top, and measure its exit speed with slow-motion video over a 0.10 m gate, as in Project 4-1-4. Use the check below to compare it with the prediction.
Use a sliding block, not a rolling marble or wheeled cart, because rolling doesn’t involve sliding friction. Friction ratios don’t depend on size, so a model’s μ should match the full-size slide’s.
🔬 Scale-Model Check
For a slope-only model test. Enter your model’s rise and slope length, the reference μ for the surface, and the exit speed you measured. The check predicts the speed and works out the μ your data implies.
✍️Written Design Memo
In your notebook or a separate document, write:
Your final design and its exit speed, with the full energy budget showing where a 40 kg rider’s and a 100 kg rider’s starting energy goes.
Why your design is safe for riders of every mass in the range. What happens to the rider’s mass in your calculation?
The tradeoffs you weighed (steeper vs. longer, grippier surface vs. a runout, and cost), and at least two tempting designs you rejected and why. (For example: “just use the grippiest surface.”)
Your friction test: what you predicted, what you measured, your percent error, and how it changed your design.
📋Rubric
Criterion
Weight
What we’re looking for
Energy Budget & Calculations
30%
Correctly budgets starting energy into exit KE and thermal energy on the slope and runout, with units and every step shown, for both a 40 kg and a 100 kg rider.
Meets the Constraints
25%
The design stays within the speed range, the budget, the angle limit, and the keep-sliding check, and the calculations prove it.
Friction Validation
20%
Tests the friction ratio of the chosen surfaces (drag tests or a scale model), calculates percent error, and updates the design if the data demands it.
Design Reasoning
15%
Explains the tradeoffs among slope length, surface, runout, and cost, rejects tempting but flawed alternatives, and explains why rider mass doesn’t change the exit speed.
Communication
10%
The memo is clear enough that someone who hasn’t seen your work could follow the reasoning and check the numbers.
📤 Turn In
Final slide design with an itemized cost, at or under the $300 budget
Energy budget and exit-speed calculations for a 40 kg and a 100 kg rider
Your friction test results (drag tests or scale model) with percent error
Written design memo with your tradeoffs and rejected designs