HRK/Chapter 2

Questions

1

The first answer depends on the definition of speed. If it is the magnitude of velocity, then speed cannot be negative, since magnitudes are never negative. When the velocity of a particle is negative, it is traveling backwards at a speed that is greater than 0.

2

To solve the rabbit problem, we need calculus, especially limits.

The question of whether he gets to the lettuce is answered by whether the time t it takes to get there is finite. We can calculate the distance traveled after time t by the simple formula, which we will derive as follows.

At time 0, the rabbit has x_0 left to go.

At time 1, the rabbit has x_0/2 left to go.

At time 2, the rabbite has x_0/2^2 left to go.

At time t, the rabbit has x_0/2^t left to go.

The position is thus:

<math>x(t) = x_0/2^t</math>

For any finite t, the answer is non-zero, and so the rabbit never gets there in finite time.

The average velocity is simply <math>{x_2 - x_1 \over t_2 - t_1} = {{x_0 \over 2^t_2} - {x_0 \over 2^t_1} \over t_2-t_1}</math>. The instantaneous velocity is <math>-\ln(2) x_0 / 2^t</math>, which tends to zero as t tends to infinity.