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Two fish swimming in a river have the following equations of motion: $$ \begin{aligned} &x_{1}=-6.4 \mathrm{~m}+(-1.2 \mathrm{~m} / \mathrm{s}) t \\ &x_{2}=1.3 \mathrm{~m}+(-2.7 \mathrm{~m} / \mathrm{s}) t \end{aligned} $$ Which fish is moving faster?

Short Answer

Expert verified
Fish 2 is moving faster.

Step by step solution

01

Identify the Velocities

Identify the velocities of both fish from their equations of motion. The velocity is represented by the coefficient of the time variable \( t \). For Fish 1, the velocity is \(-1.2 \text{ m/s}\). For Fish 2, the velocity is \(-2.7 \text{ m/s}\).
02

Recognize Direction and Speed

Recognize that both velocities are negative, indicating that both fish are moving in the same direction, presumably downstream. The speed is the absolute value of velocity. Hence, the speed of Fish 1 is \(1.2 \text{ m/s}\), and the speed of Fish 2 is \(2.7 \text{ m/s}\).
03

Compare the Speeds

Compare the speeds of the two fish. Since \(2.7 \text{ m/s} > 1.2 \text{ m/s}\), Fish 2 is moving faster than Fish 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equations of Motion
In physics, equations of motion describe how the position of an object changes over time. They often include terms such as initial position, velocity, and time, organized in a linear equation format. In our exercise, the equations for the two fish are each written as:
  • Fish 1: \( x_{1} = -6.4 \, \text{m} + (-1.2 \, \text{m/s})t \)
  • Fish 2: \( x_{2} = 1.3 \, \text{m} + (-2.7 \, \text{m/s})t \)
The equations are linear, which means they graph as straight lines, showing a constant velocity. Here, the velocity is the rate of change of position with respect to time, seen as the coefficient of \( t \):
  • For Fish 1, the velocity is \(-1.2 \, \text{m/s}\).
  • For Fish 2, it's \(-2.7 \, \text{m/s}\).
The negative sign indicates the fish are moving in a negative direction along the x-axis, which typically suggests they might be swimming downstream.
Speed Comparison
Speed is a scalar quantity representing how fast something is moving, without considering its direction. It is derived from the absolute value of velocity. In this context, both fish have velocities with negative signs, suggesting they are moving but still allowing us to determine their speed:
  • Fish 1's speed: \( \mid -1.2 \, \text{m/s} \mid = 1.2 \, \text{m/s} \)
  • Fish 2's speed: \( \mid -2.7 \, \text{m/s} \mid = 2.7 \, \text{m/s} \)
By comparing these speeds, we can determine how the two fish stack up against each other in terms of movement:
  • Fish 2 is faster because \( 2.7 \, \text{m/s} \) is greater than \( 1.2 \, \text{m/s} \).
Understanding speed in this way helps indicate how quickly each fish travels, irrespective of their direction.
Direction of Motion
The direction of motion provides context for where an object is headed. In our scenario with two fish, their equations indicate a specific movement direction. The signs of their velocities tell us which way they are swimming:A negative velocity such as \(-1.2 \, \text{m/s}\) or \(-2.7 \, \text{m/s}\) indicates the fish are moving in negative x direction.The assumption here could be that both fish swim downstream in a river where the negative direction aligns with the current.It's crucial to distinguish between directional movement and speed:
  • While both fish have the same direction (suggested by negative signs in velocity), their speeds differ.
  • This allows us to discuss motion dynamics confidently, knowing Fish 2 is not only fast but moves strongly in the designated direction.
Understanding both speed and direction ensures clear communication about how objects move and interact with their environments.

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Most popular questions from this chapter

The position-time equation of motion for a bunny hopping across a yard is $$ x_{\mathrm{f}}=8.3 \mathrm{~m}+(2.2 \mathrm{~m} / \mathrm{s}) t $$ (a) What is the initial position of the bunny? (b) What is the bunny's velocity? 36\. A bowling ball moves with constant velocity from an initial position of \(1.6 \mathrm{~m}\) to a final position of \(7.8 \mathrm{~m}\) in \(3.1 \mathrm{~s}\). (a) What is the position-time equation for the bowling ball? (b) At what time is the ball at the position \(8.6 \mathrm{~m}\) ?

Analyze You and your dog go for a walk to a nearby park. On the way your dog takes many short side trips to chase squirrels, examine fire hydrants, and so on. (a) When you arrive at the park, do you and your dog have the same displacement? Explain. (b) Have you and your dog traveled the same distance? Explain.

Two dragonflies have the following equations of motion: $$ \begin{aligned} &x_{1}=2.2 \mathrm{~m}+(0.75 \mathrm{~m} / \mathrm{s}) t \\ &x_{2}=-3.1 \mathrm{~m}+(-1.1 \mathrm{~m} / \mathrm{s}) t \end{aligned} $$ (a) Which dragonfly is moving faster? (b) Which dragonfly starts out closer to \(x=0\) at \(t=0\) ?

The equation of motion for a person riding a bicycle is \(x=6.0 \mathrm{~m}+(4.5 \mathrm{~m} / \mathrm{s}) t\). (a) Where is the bike at \(t=2.0 \mathrm{~s}\) ? (b) At what time is the bike at the location \(x=24 \mathrm{~m}\) ?

Calculate The position of a ball as a function of time is given by $$ x=3.0 \mathrm{~m}+(-5.0 \mathrm{~m} / \mathrm{s}) t $$ What is the position of the ball at \(1.5 \mathrm{~s}\) ?

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