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Chapter 10: Problem 24

What is known about the speed of an object if the angle between the velocity and acceleration vectors is (a) acute and (b) obtuse?

Short Answer

Expert verified
(a) If the angle between the velocity and acceleration vectors is acute, the speed of the object is increasing. (b) If the angle between the velocity and acceleration vectors is obtuse, the speed of the object is decreasing.

Step by step solution

01

Understand Terminology

Firstly, it is important to understand the terminologies used - speed, velocity and acceleration. Speed is a scalar quantity which refers to 'how fast an object is moving'. Velocity is a vector quantity that refers to 'the speed and direction an object is moving', whereas acceleration is 'the rate at which velocity changes', which can involve an object speeding up, slowing down, or changing direction.
02

Velocity and Acceleration Relationship - Acute Angle

In the case of an acute angle between the vectors of velocity and acceleration, it means that they are generally moving in the same direction. This implies that the speed of the object is increasing.
03

Velocity and Acceleration Relationship - Obtuse Angle

In the case of an obtuse angle between the vectors of velocity and acceleration, it means that they are generally moving in opposite directions. This implies that the speed of the object is decreasing.

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Most popular questions from this chapter

In Exercises \(27-34,\) find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j} $$

Find the angle \(\theta\) between \(r(t)\) and \(r^{\prime}(t)\) as a function of \(t .\) Use a graphing utility to graph \(\theta(t) .\) Use the graph to find any extrema of the function. Find any values of \(t\) at which the vectors are orthogonal. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j} $$

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=\frac{1}{t-1} \mathbf{i}+3 t \mathbf{j} $$

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(\theta)=(\theta-2 \sin \theta) \mathbf{i}+(1-2 \cos \theta) \mathbf{j} $$

Use the definition of the derivative to find \(\mathbf{r}^{\prime}(t)\). $$ \mathbf{r}(t)=\langle 0, \sin t, 4 t\rangle $$
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