Grandpa Ramping It Down

1 July 2023

When one is in pain, one needs distraction.

That was the case while visiting the Third Tunnel of Aggression in the DMZ north of Seoul. Walking down the 11 degree ramp, noting the distance to my destination, it occurred to me how challenging the climb out would be.

The tunnel is 358 meters in length, thankfully cool because it is also 73’ underground. Dripping water everywhere. But that ramp up is long and daunting. So I thought, what if….

Not being a math person I posed the problem to my niece, Gabrielle. “If Grandpa was in a wheelchair rolling to the bottom of the tunnel, how fast would he be traveling when he reached the bottom?”

With the help of ChatGPT and Hotpot AIs, for all those inquisitive minds, this is the answer:

Question: If 200 pound Grandpa is going down a 358 meter ramp that has an 11 degree incline, how fast will he go?

Answer: To calculate the speed at which a 200-pound person will be traveling down a 358-meter ramp with an 11-degree incline, we can use the principles of physics.

First, let’s convert the weight of the person from pounds to kilograms:

Weight of person = 200 pounds = 200 / 2.205 kg (since 1 pound is approximately equal to 0.454 kg)

Weight of person = 90.7 kg (approximately)

Now, let’s calculate the force of gravity acting on the person:

Force of gravity (Fg) = mass × gravity

Fg = 90.7 kg × 9.8 m/s²

Fg = 888.86 N (approximately)

Next, let’s calculate the component of the force of gravity acting parallel to the incline. This force is responsible for the acceleration of the person down the ramp:

Force parallel to the incline (Fpar) = Fg × sin(angle)

Fpar = 888.86 N × sin(11 degrees)

Fpar = 888.86 N × 0.191

Fpar = 169.95 N (approximately)

Now, we can use Newton’s second law of motion to calculate the acceleration (a) of the person down the ramp:

Force parallel to the incline (Fpar) = mass × acceleration

169.95 N = 90.7 kg × acceleration

Solving for acceleration:

acceleration = 169.95 N / 90.7 kg

acceleration ≈ 1.872 m/s² (approximately)

Finally, we can use the kinematic equation to calculate the final velocity (v) of the person when they reach the bottom of the ramp:

v² = u² + 2as

Since the person starts from rest (u = 0), the equation simplifies to:

v² = 2as

Plugging in the values:

v² = 2 × 1.872 m/s² × 358 m

v² = 1339.856 m²/s²

Taking the square root to find the velocity:

v ≈ √1339.856 m²/s²

v ≈ 36.61 m/s (approximately)

Question: What’s that in miles per hour?

Answer: To convert the velocity from meters per second (m/s) to miles per hour (mph), we can use the following conversion factors:

1 mile = 1609.34 meters

1 hour = 3600 seconds

Converting the velocity from m/s to mph:

v_mph = v_mps × (3600/1609.34)

Substituting the calculated velocity of 36.61 m/s:

v_mph = 36.61 m/s × (3600/1609.34)

v_mph ≈ 82.02 mph (approximately)

Therefore, the velocity of approximately 36.61 meters per second is approximately equal to 82.02 miles per hour.