Question

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

(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

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 \( x(t) = -12.1 \, \mathrm{m} + (5.2 \, \mathrm{m/s})t \).

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: \[ x(1.6) = -12.1 + 8.32 = -3.78 \, \mathrm{m} \].

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: \[ 3.0 = -12.1 + 5.2t \].

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: \[ t = \frac{15.1}{5.2} \approx 2.904 \, \mathrm{s} \].

Key concepts

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.