Question

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

Fish 2 is moving faster.

Step-by-step solution

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 \(-2.7 \text{ m/s}\).

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 \(2.7 \text{ m/s}\).

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.

Key concepts

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.