You throw a stone into some nice soft and gushy mud. It penetrates one inch, slowed by a constant force. If you wanted the stone to penetrate four inches instead of one, you would have to throw the stone into the mud?
I say eight times as fast?
No. If the retarding force is constant (which means independent of speed), and you want it to penetrate four times as far, it will have to do four times as much work against the mud. That will require four times the kinetic energy.
I think this is an unrealistic problem. Fluid resistance forces are not independent of velocity.
twice as fast
To determine how fast you would have to throw the stone into the mud in order for it to penetrate four inches instead of one, you need to understand the relationship between the force applied and the distance penetrated.
The force required to penetrate the mud is constant, meaning it does not change regardless of the distance penetrated. Therefore, if the stone penetrates four inches instead of one, it means four times the force is acting on the stone.
According to Newton's second law of motion, force (F) is equal to mass (m) multiplied by acceleration (a): F = ma.
In this case, the stone has a relatively constant mass, so the only way to increase the force acting on it is by increasing the acceleration.
Acceleration is directly related to the speed or velocity (v) of the object. The equation relating acceleration, velocity, and time is given by: a = (v - u) / t, where u is the initial velocity (which we can assume to be zero).
Simplifying the equation, we have: a = v / t.
To increase the acceleration, you need to increase the velocity. Since we are looking for the factor by which we need to increase the velocity, we can set up a proportion.
Let's say the factor by which you need to increase the velocity to penetrate four inches is x.
So, x = (new velocity) / (initial velocity).
According to the equation a = v / t, if the force is constant, the acceleration (a) can be expressed as a ratio of velocities: a = v2 / v1, where v2 is the new velocity and v1 is the initial velocity.
Since a = 4 (as we are trying to penetrate four inches instead of one), we can rewrite the equation as: 4 = v2 / v1.
Now, we need to solve for x (the factor by which we need to increase the velocity), so we rearrange the equation: x = v2 / v1 = 4.
Therefore, to penetrate four inches instead of one, you would need to throw the stone eight times as fast.