At 7:42 on a clear, dry morning, a sedan (Car A) struck the back of a compact (Car B) that was stopped at a red light on Maple Avenue, a straight, level road with a limit of 13.4 m/s (30 mph). Both drivers were hurt. The sedan’s driver says she was “going about the limit” and braked hard when she saw the light. Police photographed two skid marks, measured the crushed bumpers, and recovered the sedan’s dashcam. The city’s vehicle safety board has hired you for two jobs. First, work out how fast Car A was really going, and whether the driver’s account holds up. Second, recommend a real design change, such as a crumple-zone or bumper upgrade, that would have made this crash safer, and prove it with your own momentum analysis.
🧰Your Toolkit
Everything you need is in the unit. If a step feels shaky, review the matching activity first.
This is everything the police recovered. Assume nothing beyond what is written here.
Item
Evidence
Notes
The scene
A straight, level road, clear and dry
Treat the crash as one-dimensional.
Speed limit
13.4 m/s (30 mph)
Car A (sedan)
1,400 kg, including the driver
Struck Car B from behind.
Car B (compact)
1,000 kg, including the driver
Stopped at a red light, brakes on.
Car A’s skid before impact
14.0 m ± 0.3 m
Locked tires, ending at the point of impact.
Skid after impact
9.0 m ± 0.3 m
The bumpers interlocked and the cars skidded together to rest.
Friction ratio μ
0.70 on dry asphalt, ± 0.10 (0.40 on wet)
Locked tires. Use the same μ for all skids.
Crush damage
Car A front 0.30 m, Car B rear 0.40 m
A total of 0.70 m of crush.
Dashcam in Car A
Two roadside posts, 20.0 m ± 0.1 m apart, crossed at frames 412 and 463 of 60 fps video
Filmed before the braking began. Allow ±1 frame.
Driver’s statement
“I was going about the limit and braked hard when I saw the light.”
This is what you’re testing.
Injury benchmark
Average deceleration of 12 g or less (1 g = 9.8 m/s²)
For this problem.
🧮The Method
You can’t see Car A’s speed at the start, but you can work backward through the crash one stage at a time. Friction does the work in each skid, so use energy there. The collision is too brief for the road to matter, so use momentum there.
Stage 3, the skid after impact:v_after = √(2 μ g d₂) Stage 2, the collision:m_A v_impact = (m_A + m_B) v_after Stage 1, the skid before impact:v₀² = v_impact² + 2 μ g d₁ Dashcam:v = distance ÷ (frames ÷ fps), with δv ÷ v = 1 ÷ frames + δd ÷ d Crash pulse:KE_lost = KE_before − KE_after, F = KE_lost ÷ d_crush, a = F ÷ m
Where do these come from? In a skid, friction F_f = μmg acts over the skid length and takes away all of the kinetic energy that remains, so ½mv² = μmg·d and the mass cancels. In the collision, momentum is conserved. For the first skid, run the same idea backward: the skid removed the difference between ½mv₀² and ½mv_impact². For the crash pulse, the energy lost in the collision is absorbed by crushing metal, so F × d_crush equals it. A force on a car of mass m gives a = F ÷ m, and one g is 9.8 m/s².
Worked example: a different crash, the same method
Given
A 2,000 kg SUV rear-ends a stopped 1,200 kg hatchback. The SUV skids 10 m before impact and the pair skids 6 m after, on dry asphalt (μ = 0.70)
✓ The SUV began braking at about 18.7 m/s. The lighter hatchback felt the larger deceleration, though the force on each car was the same.
📏Constraints
Treat the road as straight and level, the crash as one-dimensional, and the two cars as locked together after impact. Assume the same friction ratio μ for every skid, and no net outside force during the collision itself.
Use the Evidence File unless you document your own measurement. Report Car A’s speeds as ranges that reflect the ±0.10 uncertainty in μ and the ±0.3 m uncertainty in each skid length.
Cross-check your reconstruction against the dashcam estimate, with its uncertainty, and say whether the two agree.
For the design, choose one option per car (or none) from the Design Options table. The total added cost across both cars may not exceed $250, and no car may gain more than 30 kg.
Safety benchmark: in a repeat of this crash, each car’s average deceleration must be 12 g or less. Check it at the fast end of your reconstructed impact-speed range, not just the central value.
Show every step with units. Every conclusion must follow from momentum and energy calculations.
🚚Deliverables
A reconstruction: Car A’s speed just after the collision, at impact, and when its driver began to brake. Give each as a range, compare the result with the dashcam and the speed limit, and say what it means for the driver’s statement.
A crash-pulse analysis: kinetic energy before and after the collision, the energy lost, the average force, and each car’s average deceleration in g, plus the change in velocity each car experienced.
A design recommendation memo to the vehicle safety board: your chosen options, a before-and-after table, why the design meets the benchmark, and what you rejected and why.
A clear statement of the uncertainty and limits of your analysis.
🗓️Suggested Pacing
Day 1: Read the evidence. Study the Evidence File, work the example by hand, and draw the crash in three stages.
Day 2: Reconstruct. Work backward through all three stages by hand for the real case, then check it in the Workbench.
Day 3: Range and cross-check. Report each speed as a range, and test your result against the dashcam and the driver’s statement.
Day 4: Crash pulse and design. Find the energy lost, the forces, and the decelerations. Use the Designer to find options that pass at the fast end of your range.
Day 5: Finalize. Write the memo, check every constraint, and state your uncertainty and limits.
🔎Reconstruct the Crash
Work the three stages by hand first. The Workbench is a check, not a substitute for showing your steps.
🔎 Reconstruction Workbench
The numbers start at the case file’s values. Work the chain by hand first, then use this to check your work and to see how much each assumption matters. Try switching the road to wet and watch what happens to the dashcam comparison.
Car A began braking at 23.6 m/s (85 km/h, 53 mph). With μ ± 0.10, the range is 21.8 to 25.2 m/s. That is 76% over the 13.4 m/s limit.
Dashcam: 20.0 m ÷ (51 ÷ 60 s) = 23.53 ± 0.58 m/s
The reconstruction and the dashcam agree within their uncertainties.
🛡️Design the Fix
Here are the upgrades the board can choose from. You may pick one for each car, or leave a car as it is.
Design option
Crush depth added
Mass added
Cost
What to know
No change
0.00 m
+0 kg
$0
The car as it is today.
Foam-filled energy-absorbing bumper beam
+0.10 m
+4 kg
$40
A modest gain in crush depth for very little mass.
Extended steel crush cans
+0.15 m
+10 kg
$70
A longer crush zone behind the bumper.
Full crumple-zone redesign
+0.30 m
+25 kg
$180
The most crush depth, at the most cost and mass.
Rigid steel bumper
-0.10 m
+30 kg
$60
Stiffer and stronger. It also leaves less room to crush.
For two cars that lock together, the energy the crash must absorb is KE_lost = [m_B ÷ (m_A + m_B)] × ½ m_A v². The same force F pushes on both cars, so the lighter car gets the larger deceleration. Notice what changes when you add mass to a car, and what changes when you add crush depth to either car.
🛡️ Crumple Zone Designer
This replays the same crash, Car A hitting stopped Car B and locking on, with the options you choose. Set the impact speed to the central or fast end of your own reconstructed range. Then check your design against every constraint.
Masses: A = 1400 kg, B = 1000 kg · total crush d = 0.70 m Momentum: Car A’s speed drops by 7.9 m/s, Car B’s rises by 11.1 m/s KE lost = [m_B ÷ (m_A + m_B)] × ½ m_A v² = 105,292 J · F = KE lost ÷ d = 150,417 N
✓ Car A’s average deceleration is 11.0 g (limit 12 g)
✗ Car B’s average deceleration is 15.3 g (limit 12 g)
✓ Total added cost is $0 (limit $250)
✓ Added mass is +0 kg on A and +0 kg on B (limit 30 kg each)
Not there yet. Adjust the options, or check a different speed.
Car A
11.0 g
Car B
15.3 g
| 12 g benchmark
✍️Written Design Memo
Address the memo to the vehicle safety board. In your notebook or a separate document, write:
What happened. Summarize the reconstruction, with ranges and the dashcam cross-check, and say what it means for the driver’s statement.
Who was hit hardest, and why. Use the changes in velocity and the decelerations. Both cars felt the same force, so why did the compact get the worse of it?
Your recommendation. The options you chose for each car, with a before-and-after table of crush depth, mass, and deceleration at the fast end of your speed range, and the cost.
Tradeoffs and rejected options. Explain at least two options you rejected and why. Address the rigid steel bumper, and any design that passes at the central speed but fails at the fast end.
Limits. What your analysis assumes (locked together, one dimension, constant μ), why an average deceleration understates the peak, and what you would test next.
📋Rubric
Criterion
Weight
What we’re looking for
Reconstruction & Calculations
30%
Works backward through all three stages with the right tool at each (energy for the skids, momentum for the collision), with units and every step shown.
Uncertainty & Cross-Check
20%
Reports speeds as ranges, handles the uncertainty in μ and the skid lengths, and tests the result against the dashcam within uncertainty.
Crash Pulse & Design Analysis
25%
Finds the energy lost, the average force and deceleration for each car, and the change in velocity of each, and checks the design at the fast end of the speed range.
Design Reasoning & Tradeoffs
15%
Explains the tradeoffs among crush depth, mass, and cost, rejects tempting but flawed options (including the rigid bumper), and connects the result to the momentum of the two cars.
Communication
10%
The memo is clear enough that a safety board member who hasn’t seen your work could follow the reasoning and check the numbers.
📤 Turn In
Your reconstruction, with all three stages, ranges, and the dashcam and speed-limit comparisons
Your crash-pulse analysis: energy lost, average force, and each car’s deceleration and change in velocity
Your design recommendation memo with its before-and-after table and cost