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Chapter 2: Problem 90

. 鈥� Calc A lizard is running in a straight line according to the following: \(x(t)=\left(0.20 \mathrm{~m} / \mathrm{s}^{3}\right) t^{3}-\left(0.40 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}-(0.65 \mathrm{~m} / \mathrm{s}) t\) (a) Determine \(v(t)\). (b) Calculate the velocity when \(t=2 \mathrm{~s}\), \(t=4 \mathrm{~s}\), and \(t=10 \mathrm{~s}\). (c) When is the lizard at rest? (d) When is the lizard moving in the positive \(x\) direction? (e) When is the lizard moving in the negative \(x\) direction? (f) When does the lizard have zero acceleration? (g) What distance does the lizard travel (not its displacement) in the first \(10 \mathrm{~s}\) ?

Short Answer

Expert verified
The velocity function is \( v(t) = 0.60 t^{2} - 0.80 t - 0.65 \) m/s. The velocities at \(t=2\) s, \(t=4\) s, and \(t=10\) s are -0.25 m/s, 1.95 m/s, and 49.35 m/s, respectively. The lizard is at rest at \(t= 0.52\) s and \(t= 2.09\) s. It moves in the positive x direction when \(0.52 < t < 2.09\), and in the negative x direction when \(0 < t < 0.52\) and \(2.09 < t < 10\). The acceleration is zero at \(t=0.67\) s. The lizard travels a total distance of 21.61 m in the first 10 seconds.

Step by step solution

01

Find Velocity Function

Differentiate the position function \(x(t)=\left(0.20 \mathrm{~m} / \mathrm{s}^{3}\right) t^{3}-\left(0.40 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}-(0.65 \mathrm{~m} / \mathrm{s}) t\). This gives the velocity function, \(v(t)\), which is \( v(t) = 0.60 t^{2} - 0.80 t - 0.65 \) m/s.
02

Calculate Velocities

The velocity at any time \(t\) can be calculated by substituting \(t\) into \(v(t)\). For \(t=2\) s, \(t=4\) s, and \(t=10\) s, we have: \(v(2) = -0.25\) m/s, \(v(4) = 1.95\) m/s, and \(v(10) = 49.35\) m/s.
03

Find Rest Times

The lizard is at rest when its velocity is zero, which is when \(v(t) = 0\). Solving this equation yields \(t= 0.52\) s and \(t= 2.09\) s.
04

Positive Movement

The lizard is moving in the positive x direction when the velocity is positive. By looking at \(v(t)\), we see this happens when \(0.52 < t < 2.09\), throughout an interval that was found from part c.
05

Negative Movement

The lizard is moving in the negative x direction when the velocity is negative. This happens when \(0 < t < 0.52\) and when \(2.09 < t < 10\), also throughout the intervals from part c.
06

Find Zero Acceleration Times

The acceleration function is the derivative of the velocity function. The acceleration is given by \(a(t) = 1.2t - 0.8\). The lizard has zero acceleration when \(a(t) = 0\), solving gives \(t=0.67\) s.
07

Calculate Travel Distance

The total distance that the lizard travels in the first 10 seconds, is the sum of the absolute values of the displacements in each interval mentioned above. So, calculate and add: \(|x(0.52) - x(0)| + |x(2.09) - x(0.52)| + |x(10) - x(2.09)|\) This yields a total distance of \(21.61\) m.

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Key Concepts

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

Position Function
The position function, often denoted as \(x(t)\), helps us understand where an object is located at any given time \(t\). In kinematics, it acts as a roadmap for the object's journey. For the lizard in our problem, its position function is \(x(t) = (0.20 \, \text{m/s}^3) t^3 - (0.40 \, \text{m/s}^2) t^2 - (0.65 \, \text{m/s}) t\). This function captures how the lizard's position changes as it moves along a straight line.

By analyzing this function, we can understand:
  • How different powers of \(t\) (like \(t^3\), \(t^2\), and \(t\)) affect the lizard's movement.
  • The contribution of each term to the overall motion - higher powers of \(t\) typically signify more complex movement patterns.
Thus, by simply plugging a specific time into \(x(t)\), we can pinpoint the exact position of the lizard at that moment.
Velocity Function
Velocity is a measure that tells us how fast and in which direction an object is moving. To find the velocity function from a position function like \(x(t)\), we use calculus, specifically differentiation.

In this problem, by differentiating \(x(t)\), we derive the velocity function \(v(t) = 0.60 t^2 - 0.80 t - 0.65\) m/s. The velocity function gives us several insights:
  • It shows the rate of change of the lizard's position over time.
  • The sign of \(v(t)\) indicates the direction of motion: positive means forward, negative means backward.
  • By plugging different times into \(v(t)\), we can determine how fast the lizard is moving at those moments.
Understanding the velocity function is crucial as it helps us not only calculate instantaneous speeds but also analyze the motion trends of the lizard.
Acceleration Function
Acceleration tells us how the velocity of an object changes with time 鈥� it's the derivative of the velocity function. Here, we find the acceleration function by differentiating \(v(t)\), which results in \(a(t) = 1.2t - 0.8\). This function gives us valuable information about the motion of the lizard:
  • The sign of \(a(t)\) indicates whether the lizard is speeding up or slowing down.
  • When \(a(t)\) is positive, the lizard accelerates; when negative, it decelerates.
  • A zero value for \(a(t)\) shows that the velocity is constant, meaning no net acceleration.
By solving \(a(t) = 0\), we can find the specific times when the lizard experiences zero acceleration, indicating periods of constant velocity. Understanding acceleration is crucial for a complete picture of how the lizard's speed and direction change over time.
Calculus in Physics
Calculus is a powerful tool in physics, particularly in kinematics, to describe motion with precision. It helps us transition between position, velocity, and acceleration with ease. Each derivative step provides deeper insight into the dynamics of the object in motion.

Here's how it applies to our exercise:
  • The first derivative of the position function provides the velocity function, detailing how position changes over time.
  • The second derivative of the position function or first derivative of velocity gives the acceleration function, detailing changes in velocity over time.
This structured approach allows physicists to predict motion, understand forces, and solve complex problems related to dynamic systems. Calculus transforms physics into a science that not only observes but also quantifies and predicts the intricate dance of motion.

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Most popular questions from this chapter

Is there any consistent reason why "up" can't be labeled as "negative" or "left" as "positive"? Explain why many physics professors and textbooks recommend choosing up and right as the positive directions in a description of motion.

The slope at a point on a position versus time graph of an object is A. the object's speed at that point. B. the object's instantaneous velocity at that point. C. the object's average velocity over a time interval centered on that point. D. the object's instantaneous acceleration at that point. E. the object's average acceleration over a time interval centered on that point.

When throwing a ball straight up, which of the following are correct about the magnitudes of its velocity \((v)\) and its acceleration \((a)\) at the highest point in its path? A. both \(v=0\) and \(a=0\) B. \(v \neq 0\), but \(a=0\) C. \(v=0\), but \(a=9.8 \mathrm{~m} / \mathrm{s}^{2}\) D. \(v \neq 0\), but \(a=9.8 \mathrm{~m} / \mathrm{s}^{2}\) E. There is not enough information to determine the velocity \((v)\) and acceleration \((a)\).

\- Wes and Lindsay stand on the roof of a building. Wes leans over the edge and drops an apple. Lindsay waits \(1.25 \mathrm{~s}\) after Wes releases his fruit and throws an orange straight down at \(28 \mathrm{~m} / \mathrm{s}\). Both pieces of fruit hit the ground simultaneously. Calculate the common height from which the fruits were released. Ignore the effects of air resistance.

A car traveling \(80 \mathrm{~km} / \mathrm{h}\) is \(1500 \mathrm{~m}\) behind a truck traveling at \(70 \mathrm{~km} / \mathrm{h}\). How long will it take the car to reach the truck?
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