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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

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=\frac{2 t}{8+t^{3}} \mathbf{i}+\frac{2 t^{2}}{8+t^{3}} \mathbf{j} $$

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

A projectile is launched with an initial velocity of 100 feet per second at a height of 5 feet and at an angle of \(30^{\circ}\) with the horizontal. (a) Determine the vector-valued function for the path of the projectile. (b) Use a graphing utility to graph the path and approximate the maximum height and range of the projectile. (c) Find \(\mathbf{v}(t),\|\mathbf{v}(t)\|,\) and \(\mathbf{a}(t)\) (d) Use a graphing utility to complete the table. $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{t} & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 \\ \hline \text { Speed } & & & & & & \\ \hline \end{array} $$ (e) Use a graphing utility to graph the scalar functions \(a_{\mathbf{T}}\) and \(a_{\mathrm{N}} .\) How is the speed of the projectile changing when \(a_{\mathrm{T}}\) and \(a_{\mathbf{N}}\) have opposite signs?

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Prove that the vector \(\mathbf{T}^{\prime}(t)\) is \(\mathbf{0}\) for an object moving in a straight line.

Consider a particle moving on a circular path of radius \(b\) described by $$ \begin{aligned} &\mathbf{r}(t)=b \cos \omega t \mathbf{i}+b \sin \omega t \mathbf{j}\\\ &\text { where } \omega=d \theta / d t \text { is the constant angular velocity. } \end{aligned} $$ Find the acceleration vector and show that its direction is always toward the center of the circle.
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