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Solved Problem on Kinematics
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Two boats depart from the same point and travel on the same straight line, with constant speeds of 25 km/h and 35 km/h. Communication between the two boats is possible by radio as long as the distance between them does not exceed 600 km. Find the time interval during which the two boats can communicate, in the following cases:
a) the two boats move in the same direction;
b) the slower boat departs two hours before the other and moves in the same direction;
c) The two boats depart at the same time and move in opposite directions.


Problem Data Problem diagram

We choose a reference frame with the positive axis pointed to the right (figure 1).

Two boats are departing from the same point, and in the same direction, with speeds v1 equal to 25 kilometers per hour and v2 equal to 35 kilometers per hour.
figure 1

Let us consider that the point from which the boats depart is the origin of the reference frame S01 = S02 = 0

Solution

a) The boats move at constant speeds, the equations of displacement as a function of time is
\[ \bbox[#99CCFF,10px] {S=S_{0}+vt} \]
writing this equation for the two boats, we have
\[ \begin{align} S_{1}=S_{01}+v_{1}t\\ S_{1}=0+25t \;\,\\ S_{1}=25t \quad\; \tag{I-a} \end{align} \]
\[ \begin{align} S_{2}=S_{02}+v_{2}t\\ S_{2}=0+35t \;\,\\ S_{2}=35t \quad\; \tag{I-b} \end{align} \]
The faster boat 2 moves away from the boat 1 until the distance between then is greater than 600 km, and communication is no longer possible (figure 2).

Boat 2, with a speed of 35 kilometers per hour, moving away from boat 1, with a speed of 25 kilometers per hour.
figure 2

Calculating the difference between the two expressions (I-a) and (I-b), we have
\[ \frac{\begin{matrix}S_{2}=35t\\(\text{-})\ \ \ S_{1}=25t\ \ \ \ \ \ \ \end{matrix}}{S_{2}-S_{1}=35t-25t} \]
We have Δ S = S2 - S1 = 600 km
\[ \Delta S=10t\\ 10t=600\\ t=\frac{600}{10} \]
\[ \bbox[#FFCCCC,10px] {t=60\;\text{h}} \]

b) Boat 1 departs and sails for 2 hours away from the starting point while boat 2 remains stationary. During this time the communication between the two boats is possible. When boat 1 reaches a position S1, boat 2 starts to navigate from the origin and navigates until it overtakes boat 1, moving away until the distance between them is greater than 600 km and communication is no longer possible (figure 2).

Image 1 - boat 1 sails for 2 hours, with a speed of 25 kilometers per hour, reaching position S1, when boat 2 departs from the origin with speed of 35 kilometers per hour; image 2 - boat 1 reaches and overtakes boat 2; image 3 - boat 2 reaches a position S2 which is 600 kilometers ahead of boat 1.
figure 3

Since t is the travel time of boat 2, the travel time of boat 1 will be t+2 indicating that it has two more hours of travel by writing the equations of motion we have
\[ \begin{align} S_{1}=S_{01}+v_{1}(\;t+2\;)\\ S_{1}=0+25(\;t+2\;) \;\,\\ S_{1}=25(\;t+2\;) \quad\; \tag{II-a} \end{align} \]
\[ \begin{align} S_{2}=S_{02}+v_{2}t\\ S_{2}=0+35t\;\;\\ S_{2}=35t \quad\; \tag{II-b} \end{align} \]
Calculating the difference between the two expressions (II-a) and (II-b), we have
\[ \frac{\begin{matrix}S_{2}=35t\\(\text{-})\ \ \ S_{1}=25(\;t+2\;)\ \ \ \ \ \;\end{matrix}}{S_{2}-S_{1}=35t-25(\;t+2\;)}\\ \Delta S=35t-25t-50\\ 10t=600+50\\ t=\frac{650}{10} \]
\[ \bbox[#FFCCCC,10px] {t=65\;\text{h}} \]

c) Let us assume that boat 1 departs in the opposite direction to the frame orientation, thus its velocity will be negative (v1=−25 km/h - figure 4).

Boats are sailing in opposite directions from the origin. Boat 1 with a speed of -25 kilometers per hour and boat 2 with a speed of 35 kilometers per hour.
figure 4

The equations for both boats will be
\[ \begin{align} S_{1}=S_{01}+v_{1}t\\ S_{1}=0-25t \;\,\\ S_{1}=-25t \quad\; \tag{III-a} \end{align} \]
\[ \begin{align} S_{2}=S_{02}+v_{2}t\\ S_{2}=0+35t \;\,\\ S_{2}=35t \quad\; \tag{III-b} \end{align} \]
Calculating the difference between the two expressions (III-a) and (III-b), we have
\[ \frac{\begin{matrix}S_{2}=35t\ \ \ \\ (\text{-})\ \ \ S_{1}=-25t\ \ \ \ \ \ \ \end{matrix}}{S_{2}-S_{1}=35t+25t}\\ \Delta S\;=\;60t\\ 60t\;=\;600\\ t\;=\;\frac{600}{60} \]
\[ \bbox[#FFCCCC,10px] {t\;=\;10\;\text{h}} \]

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