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Chapter 3: Problem 6

An object moving along a line has a constant negative acceleration. Describe the velocity of the object.

Short Answer

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Question: Describe how the velocity of an object moving along a line with constant negative acceleration will behave over time. Answer: The velocity of the object depends on its initial velocity and the magnitude of the negative acceleration. If the object starts with a positive velocity, the negative acceleration will decrease its velocity until it stops, and then it will reverse its direction. If the object starts with a negative velocity, its velocity will continually become more negative due to the negative acceleration. In both cases, the object eventually moves in the negative direction.

Step by step solution

01

Understanding the relationship between acceleration and velocity

Acceleration is the rate of change of velocity with respect to time. Mathematically, it can be represented as: a = \frac{dv}{dt} Where a is the acceleration, dv is the change in velocity, and dt is the time interval.
02

Interpret negative acceleration

Negative acceleration means that the acceleration is acting in the opposite direction of the motion of the object. This means that the object is either slowing down as it moves in the positive direction, or speeding up as it moves in the negative direction.
03

Analyze constant acceleration

Constant acceleration means that the magnitude and direction of the acceleration remain the same during the entire motion of the object. Mathematically, this signifies that the acceleration function is a(t) = a_0 where a_0 is a constant value and a(t) represents the acceleration at any time t. Now, integrate the constant acceleration function with respect to time to obtain the velocity function: v(t) = \int a(t) dt = \int a_0 dt = a_0t + C where v(t) is the velocity function at any time t and C is the integration constant, which represents the initial velocity of the object (v_0).
04

Describe the velocity of the object

Considering that the acceleration is negative (a_0 < 0), the velocity function can be written as: v(t) = v_0 - a_0t The velocity of the object depends on the initial velocity (v_0) and the magnitude of the negative acceleration (a_0). If the object starts with a positive velocity, the negative acceleration will decrease its velocity until it stops (velocity becomes zero) and then reverses its direction. If the object starts with a negative velocity, its velocity will continually become more negative due to the negative acceleration. In both cases, the object eventually moves in the negative direction.

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Most popular questions from this chapter

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}(f(x) g(x))\right|_{x=1}$$

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Suppose \(f(2)=2\) and \(f^{\prime}(2)=3 .\) Let $$g(x)=x^{2} \cdot f(x) \text { and } h(x)=\frac{f(x)}{x-3}$$ a. Find an equation of the line tangent to \(y=g(x)\) at \(x=2\) b. Find an equation of the line tangent to \(y=h(x)\) at \(x=2\)

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