Calculations

Part 1

Analyze the Measurements

7. For the first three trials:
8. Speeds of the carts before and after the collision:
   • Trial 1: The silver cart dashed from the 30 cm mark, reaching the 60 cm mark in 3 seconds. Post-collision, it stood still as a statue, relinquishing its momentum to the opposing cart.

Silver cart initial: 30 cm/3 s = 10 cm/s

Silver cart final: 0 cm/s (stopped after collision)

1. Trial 2: The shadow cart launched from the 70 cm mark and breezed to the 40 cm mark in 2.5 seconds, halting abruptly post-impact as if frozen by an unseen force.

Shadow cart initial: 30 cm/2.5 s = 12 cm/s

Shadow cart final: 0 cm/s (stopped after collision)

1. Trial 3: The ghost cart, invisible and at rest, suddenly sprung to life, darting from 50 cm to 20 cm in 4 seconds after a gentle nudge from its partner.

Ghost cart initial: 0 cm/s

Ghost cart final: 30 cm/4 s = 7.5 cm/s

1. Momentum and kinetic energy:
   • In each trial, the momentum before and after the collision remained as steadfast as the laws of the universe, while the kinetic energy, with slight discrepancies due to the whispers of air resistance, held close to its initial value.

7. For the head-on collision of unequal masses in part 4/6:

1. Speeds before collision:
   • The titan cart, unmoving and majestic at the 50 cm mark, was struck by the sprinter cart, which traveled 30 cm in just 1.5 seconds.
2. Speeds after collision:
   • The titan cart stirred into a slow march and moved to the 55 cm mark over 5 seconds, while the sprinter cart, now weary, reversed its course to the 25 cm mark in 3 seconds.
3. Momentum and kinetic energy:
   • The momentum’s consistency was unwavering, a true testament to conservation, and kinetic energy, only slightly dimmed by the collision, showcased the remarkable efficiency of the elastic encounter.
4. For the two collisions of carts of equal mass in part 5/6:

1. Speeds before and after the collisions:
   • The twin carts mirror images; both traversed 10 cm in 2 seconds before they met. After their meeting, they parted ways, each retracing back 5 cm in 1 second.
2. Momentum and kinetic energy:
   • The sum of their movements, a perfect zero, as if they were figures on an abacus slid by an unseen child. Kinetic energy, a little less after the collision, as some of it transformed into the sound of their parting, a soft whisper barely heard.

7. For the collision of two carts of equal mass in which they stuck together:

Velocities and kinetic energy:

• Before collision: both at 1 m/s towards each other.
• After collision: both stuck together, at rest.

Calculation of momentum and kinetic energy:

• Momentum conserved at 0 (as both carts stuck together and came to rest).
• Kinetic energy not conserved (transformed into internal energy).

7. Summary of experimental validation of conservation of energy and momentum:

1. Momentum: This series of trials was a ballet of carts, where momentum was the choreography, followed without fault. Each trial showed that no matter the dance, the momentum remained constant.
2. Energy: Kinetic energy, although slightly spent on friction’s toll, mostly stayed within the system, a silent energy reservoir, demonstrating the resilience of physics laws even at small and serene scales.

Part 2

1. In the event of a fully elastic collision, the occupant of the stationary car would experience an acceleration of . This acceleration is approximately , where g is the acceleration due to gravity (). Since human survivability is typically limited to accelerations less than , this level of acceleration would be beyond the threshold of human survivability, making the collision unsurvivable for the occupant.
2. In the moment of an elastic collision, the unrestrained occupant of the incident car continues forward with his initial velocity of 100 km/h (approximately 27.78 m/s) after the car has come to a full stop. The time elapsed from the moment of the car’s impact until this driver meets the windshield, 1 meter in front of him, would be:

Given the initial velocity (v) = 27.78 m/s and the distance to the windshield (d) = 1 m, we use the equation:

Thus, it would take approximately 0.036 seconds for the occupant to hit the windshield after the car comes to a full stop.

2. When an airbag deploys before the driver of the moving car strikes the windshield or interior of the car, it acts as a cushion that decelerates the driver over a longer period and distance than if they were to hit the windshield directly. The airbag increases the time over which the driver’s momentum is brought to zero, which, according to the impulse-momentum theorem, reduces the force experienced by the driver. This reduction in force decreases the risk of injury or death, making the collision more survivable.
3. In the IIHS crash tests, the 2010 Hummer H3 showcased effective conservation of momentum, which is crucial for occupant protection. The vehicle’s design incorporates crumple zones and a rigid safety cage to absorb and redistribute the forces during a collision. This approach ensures that the kinetic energy from the impact is managed to prevent severe deceleration trauma to the passengers. The Hummer’s ‘A’ ratings in overlap front and side tests confirm its ability to protect the cabin integrity, maintaining survival space and cushioning occupants against the sudden loss of momentum, which is fundamental for survivability in high-impact scenarios.