## Wednesday, April 05, 2023

### ChatGPT and Projectile Motion, Part 2

If you missed my previous physics questions to ChatGPT, check them out here and here

In this particular case, I asked ChatGPT a rather common question, and ended up having an argument with it. My questions are in red while ChatGPT's responses are in blue.

Three projectiles of the same mass are launched from the top of a cliff with the same initial speed. The first projectile is launched at an angle of 20 degrees above the horizontal. The second is launched horizontally. The third is launched at an angle 20 degrees below the horizontal. Which projectile hit the ground with the largest speed?

All three projectiles are launched with the same initial speed, which means they all have the same initial kinetic energy. The only difference is the angle at which they are launched.

The maximum range of a projectile launched at a given speed is achieved when it is launched at an angle of 45 degrees above the horizontal. This means that the first and third projectiles will have the same range, but the second projectile, launched horizontally, will have a shorter range.
However, the question asks which projectile hits the ground with the largest speed, not which has the greatest range. To answer this question, we need to consider the vertical component of each projectile's velocity at impact.

The first projectile is launched at an angle of 20 degrees above the horizontal, so it will have a significant vertical component of velocity at impact, causing it to hit the ground with a larger speed than the third projectile, which is launched at an angle of 20 degrees below the horizontal and will have a smaller vertical component of velocity at impact.

The second projectile, launched horizontally, will have no vertical component of velocity at impact, so it will hit the ground with the smallest speed of the three projectiles.

Therefore, the first projectile, launched at an angle of 20 degrees above the horizontal, will hit the ground with the largest speed.

It started off so well, realizing that all three have the same KE. But then, after it answered something that wasn't asked (the range), it went on to be obsessed with the vertical component of the velocity.

I decided to challenge its answer.

But they all have the same initial energy in the beginning. Shouldn't they have the same energy at the bottom when they hit the ground?

You're correct that all three projectiles have the same initial kinetic energy, and conservation of energy tells us that the total energy of the system must be conserved throughout the motion. Therefore, at the bottom of the cliff, all three projectiles will have the same total energy as they did at the top of the cliff.

However, the question specifically asks which projectile hits the ground with the largest speed, not which has the most kinetic energy. The speed of a projectile at any point during its motion depends not only on its kinetic energy but also on its direction and the effects of external forces such as gravity and air resistance.

In the case of these three projectiles, they all have different launch angles, which means they will follow different trajectories and experience different vertical and horizontal components of acceleration due to gravity. These differences will cause the projectiles to hit the ground with different speeds, even though they all have the same initial kinetic energy.

It's a bit puzzling what it is thinking at this point. If they have the same KE in the end, via conservation of energy, then should they also have the same speed?

But kinetic energy is 1/2*m*v^2. If they have the same kinetic energy, they should have the same speed.

Yes, you are correct that the kinetic energy of an object is given by the formula KE = 1/2 * m * v^2, where m is the mass of the object and v is its velocity. If three projectiles of the same mass are launched from the top of a cliff with the same initial speed, they will all have the same initial kinetic energy, and therefore the same total energy throughout their motion.

However, as the projectiles follow their different trajectories, they will experience different amounts of air resistance and vertical and horizontal components of acceleration due to gravity. These differences in the forces acting on the projectiles will affect their velocities as they reach the ground.
The projectile launched horizontally will experience no vertical acceleration due to gravity, so it will not gain any speed in the vertical direction as it falls, only in the horizontal direction. The first and third projectiles, on the other hand, will experience significant vertical accelerations due to gravity, which will affect their speeds at impact.

Therefore, even though all three projectiles have the same initial kinetic energy, the projectile launched at an angle of 20 degrees above the horizontal will hit the ground with the largest speed due to the combination of its initial horizontal velocity and the vertical velocity gained as it falls.

How can the projectile launched horizontally not experience vertically acceleration due to gravity? After all, it is falling to the ground!

You are correct that a projectile launched horizontally will experience vertical acceleration due to gravity, and it will also fall to the ground just like the other two projectiles.

What I meant to say is that the horizontal component of the velocity of the projectile launched horizontally will remain constant throughout its motion, since there are no external forces acting on it in the horizontal direction. This means that the horizontal velocity of the projectile will be the same at the top of the cliff and at the moment it hits the ground.

However, the vertical component of the velocity of the projectile launched horizontally will increase due to the acceleration of gravity. At the top of the cliff, the vertical component of the velocity is zero, but as the projectile falls, the vertical velocity will increase, and the projectile will hit the ground with a non-zero vertical speed.

Therefore, while the horizontal velocity of the horizontally launched projectile remains constant, the vertical component of its velocity will increase due to gravity, resulting in a non-zero speed at impact. However, since the horizontal velocity of the projectile remains constant, it will hit the ground with the smallest speed of the three projectiles.

The question asks about the speed, not the velocity. All three gain in speed as it falls to the ground. If their kinetic energy is the same, their speeds must be the same.

You are correct that the question asks about the speed of the projectiles, not the velocity. The speed of an object is the magnitude of its velocity, and it is calculated by taking the absolute value of its velocity vector.

If all three projectiles have the same initial kinetic energy, then they must have the same initial speed. As the projectiles fall, they will gain speed due to the force of gravity.

Therefore, at the moment of impact, all three projectiles will have the same kinetic energy, which means they will have the same speed. The angle at which the projectiles are launched and the resulting trajectories they follow may affect their final velocities, but their final speeds will be the same since they all have the same kinetic energy at impact.

So, to answer the original question, all three projectiles will hit the ground with the same speed, even though they were launched at different angles.

Phew!! That felt like pulling teeth!

It was as if it doesn't know the difference between speed and velocity, and I'm wondering if it was arguing with me about the vertical component of the velocity. It still makes no sense because clearly I explicitly mentioned "speed" at the very beginning.

Once again, if you don't know what you are looking for, there is no guarantee that you'll get the correct answer and consistent explanation.

Zz.