- #1
greypilgrim
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- 36
Hi.
An ant walks with 1 cm/s along a rubber rope with initial length 10 cm that is stretched uniformly with 10 cm/s. Find its velocity at time ##t##.
This is a a well-known problem, but I'd like to know what's wrong with the following alternative approach:
At time ##t>0##, any point ##x_0## on the rope gets stretched to
$$x(t)=x_0\cdot (1+\frac{10\,\frac{\text{cm}}{\text{s}}}{10\,\text{cm}}\cdot t)=x_0\cdot (1+\frac{1}{\text{s}}\cdot t)\enspace.$$
This is equivalent to
$$x_0=\frac{x}{1+\frac{1}{\text{s}}\cdot t}$$
The given speed of the ant is with respect to the rope at any time ##t##, so ##\frac{dx}{dt}=1\,\frac{\text{cm}}{\text{s}}##. However, we want its velocity with respect to a fixed scale, such as the rubber band at ##t=0##. So I get
$$v(t)=\frac{dx_0}{dt}=\frac{dx_0}{dx}\cdot \frac{dx}{dt}=\frac{1}{1+\frac{1}{\text{s}}\cdot t}\cdot 1\,\frac{\text{cm}}{\text{s}}\enspace.$$
This of course is terribly wrong, as it would mean that the ant is even slower than 1 cm/s seen from the outside. But what exactly is wrong?
An ant walks with 1 cm/s along a rubber rope with initial length 10 cm that is stretched uniformly with 10 cm/s. Find its velocity at time ##t##.
This is a a well-known problem, but I'd like to know what's wrong with the following alternative approach:
At time ##t>0##, any point ##x_0## on the rope gets stretched to
$$x(t)=x_0\cdot (1+\frac{10\,\frac{\text{cm}}{\text{s}}}{10\,\text{cm}}\cdot t)=x_0\cdot (1+\frac{1}{\text{s}}\cdot t)\enspace.$$
This is equivalent to
$$x_0=\frac{x}{1+\frac{1}{\text{s}}\cdot t}$$
The given speed of the ant is with respect to the rope at any time ##t##, so ##\frac{dx}{dt}=1\,\frac{\text{cm}}{\text{s}}##. However, we want its velocity with respect to a fixed scale, such as the rubber band at ##t=0##. So I get
$$v(t)=\frac{dx_0}{dt}=\frac{dx_0}{dx}\cdot \frac{dx}{dt}=\frac{1}{1+\frac{1}{\text{s}}\cdot t}\cdot 1\,\frac{\text{cm}}{\text{s}}\enspace.$$
This of course is terribly wrong, as it would mean that the ant is even slower than 1 cm/s seen from the outside. But what exactly is wrong?
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