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Chapter 6: Problem 28

Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. $$a(t)=-32, v(0)=50, s(0)=0$$

Short Answer

Expert verified
Answer: The expressions for the position and velocity are: Position function: $$s(t) = -16t^2 + 50t$$ Velocity function: $$v(t) = -32t + 50$$

Step by step solution

01

Integrate the acceleration function to find the velocity function

To find the velocity function, integrate the given acceleration function, $$a(t) = -32$$, with respect to time: $$v(t) = \int a(t)\, dt = \int -32 \, dt.$$ Since the integral of a constant is just the constant times the variable, we have $$v(t) = -32t + C_1$$ We're given the initial velocity $$v(0) = 50$$, which we can use to find the value of $$C_1$$: $$50 = -32(0) + C_1$$ $$C_1 = 50$$ So, the velocity function is $$v(t) = -32t + 50$$
02

Integrate the velocity function to find the position function

Next, we'll find the position function by integrating the velocity function with respect to time: $$s(t) = \int v(t) \, dt = \int (-32t + 50) \, dt$$ Integrating, we get $$s(t) = -16t^2 + 50t + C_2$$ Now, use the initial position $$s(0) = 0$$ to find $$C_2$$: $$0 = -16(0)^2 + 50(0) + C_2$$ $$C_2 = 0$$ So, the position function is $$s(t) = -16t^2 + 50t$$ The expressions for the position and velocity of the object moving along a straight line are as follows: Position function: $$s(t) = -16t^2 + 50t$$ Velocity function: $$v(t) = -32t + 50$$

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Key Concepts

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

Understanding Acceleration
Acceleration is a key concept in kinematics. It refers to the rate at which an object's velocity changes over time. In the exercise, the acceleration function is given as a constant: \[a(t) = -32\]. This means that the object is speeding up or slowing down at a consistent rate of \(-32\) units per second squared.
  • When the acceleration is constant, like in this example, it can easily be integrated to find the velocity function.
  • The negative sign indicates a reduction in the velocity over time, assuming positive initial velocity.
Integrating the acceleration allows us to uncover more about the motion of the object, as it helps derive future velocity states.
Exploring Velocity
Velocity describes the speed of an object in a given direction. It's a vector quantity, meaning it has both magnitude and direction. In our example, the velocity function is found by integrating the acceleration function:\[v(t) = -32t + 50\]. This equation tells us how fast the object is moving and in which direction as time progresses.
  • The term \(-32t\) is derived from integrating the acceleration, representing the change in velocity over time.
  • The constant \(50\) symbolizes the initial velocity, or the speed at time \(t = 0\).
As time increases, the \(-32t\) term decreases the velocity due to the negative acceleration, showcasing how velocity changes due to the effects of acceleration.
Understanding the Position Function
The position function gives the location of the object at any point in time. It's obtained by integrating the velocity function. For this exercise, we have:\[s(t) = -16t^2 + 50t\]. This function projects where the object will be as time advances.
  • The \(-16t^2\) term results from a successive integration, indicating an increase in the effect of negative acceleration over time on position.
  • The \(50t\) term ensures that the initial velocity is factored into the position calculations.
Analyzing the position function is crucial because it consolidates all the motion details, showing how initial conditions and acceleration influence overall displacement.

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Most popular questions from this chapter

The U.S. government reports the rate of inflation (as measured by the Consumer Price Index) both monthly and annually. Suppose that for a particular month, the monthly rate of inflation is reported as \(0.8 \%\). Assuming that this rate remains constant, what is the corresponding annual rate of inflation? Is the annual rate 12 times the monthly rate? Explain.

Find the volume of the solid generated in the following situations. The region \(R\) in the first quadrant bounded by the graphs of \(y=x\) and \(y=1+\frac{x}{2}\) is revolved about the line \(y=3\).

A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of \(5 \mathrm{kg} / \mathrm{m}\). a. How much work is required to wind the entire chain onto the cylinder using the winch? b. How much work is required to wind the chain onto the cylinder if a \(50-\mathrm{kg}\) block is attached to the end of the chain?

Determine whether the following statements are true and give an explanation or counterexample. a. \(\frac{d}{d x}(\sinh \ln 3)=\frac{\cosh \ln 3}{3}.\) b. \(\frac{d}{d x}(\sinh x)=\cosh x\) and \(\frac{d}{d x}(\cosh x)=-\sinh x.\) c. Differentiating the velocity equation for an ocean wave \(v=\sqrt{\frac{g \lambda}{2 \pi} \tanh \left(\frac{2 \pi d}{\lambda}\right)}\) results in the acceleration of the wave. d. \(\ln (1+\sqrt{2})=-\ln (-1+\sqrt{2}).\) e. \(\int_{0}^{1} \frac{d x}{4-x^{2}}=\frac{1}{2}\left(\operatorname{coth}^{-1} \frac{1}{2}-\cot ^{-1} 0\right).\)

Determine whether the following statements are true and give an explanation or counterexample. a. When using the shell method, the axis of the cylindrical shells is parallel to the axis of revolution. b. If a region is revolved about the \(y\) -axis, then the shell method must be used. c. If a region is revolved about the \(x\) -axis, then in principle, it is possible to use the disk/washer method and integrate with respect to \(x\) or the shell method and integrate with respect to \(y\)
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