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Cleo the black lab runs to pick up a stick on the ground at the location \(x=3.0 \mathrm{~m}\). The equation of motion for Cleo is \(x=-12.1 \mathrm{~m}+(5.2 \mathrm{~m} / \mathrm{s})\) t. (a) Where is Cleo at \(t=1.6 \mathrm{~s}\) ? (b) At what time does Cleo reach the stick?

Short Answer

Expert verified
(a) Cleo is at -3.78 m at t=1.6 s. (b) Cleo reaches the stick at t≈2.904 s.

Step by step solution

01

Identify Given Variables

The initial position of Cleo is given as \( x_0 = -12.1 \, \mathrm{m} \). The velocity \( v = 5.2 \, \mathrm{m/s} \). The position function of Cleo is \( x(t) = -12.1 \, \mathrm{m} + (5.2 \, \mathrm{m/s})t \).
02

Calculate Cleo's Position at t=1.6 s

Substitute \( t = 1.6 \, \mathrm{s} \) into the position equation: \[ x(1.6) = -12.1 + (5.2)(1.6) \] Calculate this to find Cleo's position: \[ x(1.6) = -12.1 + 8.32 = -3.78 \, \mathrm{m} \].
03

Set Up the Equation to Find the Time Cleo Reaches the Stick

To find the time \( t \) when Cleo reaches \( x = 3.0 \, \mathrm{m} \), set the position equation equal to 3.0 m: \[ 3.0 = -12.1 + 5.2t \].
04

Solve for t When Cleo Reaches the Stick

Rearrange the equation from Step 3 to solve for \( t \): \[ 3.0 = -12.1 + 5.2t \] First, add 12.1 to both sides: \[ 15.1 = 5.2t \] Divide both sides by 5.2: \[ t = \frac{15.1}{5.2} \approx 2.904 \, \mathrm{s} \].

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Key Concepts

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

Equation of Motion
Understanding the equation of motion is fundamental in kinematics problems. It's like a map that tells us where an object is at any given time. The general form of a linear equation of motion is given as:\[ x(t) = x_0 + vt \] where:
  • \( x(t) \) is the position at time \( t \).
  • \( x_0 \) is the initial position.
  • \( v \) is the velocity.
  • \( t \) is the time elapsed.
In this case, the equation of motion for Cleo is \( x(t) = -12.1 \, \mathrm{m} + (5.2 \, \mathrm{m/s})t \). This form means Cleo starts at -12.1 meters and moves with a velocity of 5.2 meters per second. Just remember, by plugging in a specific \( t \) value, you get the position of Cleo at that time!
Velocity
Velocity is not just about speed; it also includes direction. It's a vector quantity. In one-dimensional kinematics, as in Cleo's case, you denote velocity with just a number, but it represents how fast and in which direction Cleo is moving.Cleo's velocity is given as \( 5.2 \, \mathrm{m/s} \). The positive sign indicates she's moving in the positive direction of the axis. Every second, Cleo covers 5.2 meters. It's like Cleo is on a time-driven conveyor belt where every tick of the clock propels her further!Here's a trick: Think of velocity as the slope of Cleo's journey on a graph of position vs. time. A steeper slope means a higher velocity.
Position
Position tells you exactly where Cleo is on the imaginary line stretching from the starting point towards her final destination. Initially, she starts at \( -12.1 \, \mathrm{m} \). As time progresses, her position changes because of her velocity. To find Cleo's position at any given time, use her position equation: \( x(t) = -12.1 + (5.2)t \). For example, when \( t = 1.6 \, \mathrm{s} \), substituting in the equation gives us \( x(1.6) = -3.78 \, \mathrm{m} \). This means that 1.6 seconds into her run, she has moved to -3.78 meters."
Time Calculation
Time calculation often involves determining when an object reaches a specific point. For Cleo, we want to know the time \( t \) when she reaches \( x = 3.0 \, \mathrm{m} \).This can be done by rearranging her position equation for \( t \): \[ 3.0 = -12.1 + 5.2t \]First, isolate \( t \) by adding 12.1 to both sides, resulting in:\[ 15.1 = 5.2t \]Then, solve for \( t \) by dividing both sides by 5.2:\[ t = \frac{15.1}{5.2} \approx 2.904 \, \mathrm{s} \]This means Cleo will reach the location of the stick after approximately 2.904 seconds. Understanding time calculation allows you to predict when events occur during motion.

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