Question

problem 2: a boat travels in a lake. it makes a displacement of (1200, 760) m in a time of 2 minutes. find the velocity of the boat in meters per second.

Explanation:

Step1: Calculate the magnitude of displacement

The displacement vector is \((1200, 760)\) m. The magnitude of displacement \(d\) is given by the Pythagorean theorem: \(d=\sqrt{x^{2}+y^{2}}\), where \(x = 1200\) m and \(y = 760\) m.
\[

$$\begin{align*} d&=\sqrt{1200^{2}+760^{2}}\\ &=\sqrt{1440000 + 577600}\\ &=\sqrt{2017600}\\ & = 1420.42\space\text{(approx)} \end{align*}$$

\]

Step2: Convert time to seconds

The time is 2 minutes. Since 1 minute = 60 seconds, \(t=2\times60 = 120\) seconds.

Step3: Calculate velocity

Velocity \(v\) is given by the formula \(v=\frac{d}{t}\).
\[
v=\frac{1420.42}{120}\approx11.84\space\text{m/s}
\]
(Alternatively, we can also calculate the components of velocity first. The x - component of velocity \(v_x=\frac{1200}{120}=10\) m/s, the y - component of velocity \(v_y=\frac{760}{120}\approx6.33\) m/s. Then the magnitude of velocity \(v=\sqrt{v_x^{2}+v_y^{2}}=\sqrt{10^{2}+6.33^{2}}=\sqrt{100 + 40.07}=\sqrt{140.07}\approx11.83\) m/s (the slight difference is due to rounding in intermediate steps).)

Answer:

The velocity of the boat is approximately \(11.84\space\text{m/s}\) (or using the component method, the magnitude of velocity is approximately \(11.83\space\text{m/s}\)). If we consider the exact calculation from the displacement magnitude:
First, \(d=\sqrt{1200^{2}+760^{2}}=\sqrt{1440000 + 577600}=\sqrt{2017600}=4\sqrt{126100}=4\times355.1\approx1420.4\) (more accurately, \(\sqrt{2017600}=1420.42246\)). Then \(v = \frac{1420.42246}{120}\approx11.84\space\text{m/s}\). So the velocity is approximately \(\boldsymbol{11.84\space\text{m/s}}\) (or if we use the component - based velocity magnitude calculation, approximately \(\boldsymbol{11.83\space\text{m/s}}\)).