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Chapter 3: Problem 3

A ball is thrown at an angle between \(0^{\circ}\) and \(90^{\circ}\) with respect to the horizontal. Its velocity and acceleration vectors are parallel to each other at a) \(0^{\circ}\) c) \(60^{\circ}\) e) none of the b) \(45^{\circ}\) d) \(90^{\circ}\) above

Short Answer

Expert verified
Answer: e) None of the angles

Step by step solution

01

Understand and identify relevant parameters

In this problem, we are given the projectile motion of a ball thrown at an angle with respect to the horizontal. The relevant parameters are the velocity components (\(v_x\) and \(v_y\)), and the acceleration components (\(a_x\) and \(a_y\)).
02

Identify the conditions for parallel vectors

For the velocity and acceleration vectors to be parallel, their horizontal and vertical components must be proportional. That is, the ratio of \(v_x\) to \(a_x\) equals the ratio of \(v_y\) to \(a_y\).
03

Determine the acceleration components

Since the only acceleration acting on the ball is gravity (which acts in the vertical direction), we have \(a_x=0\) and \(a_y=-g\) (with \(g\) being the acceleration due to gravity).
04

Determine the velocity components

The horizontal and vertical components of the velocity can be expressed in terms of the initial velocity \(v_0\) and the launch angle \(\theta\): \(v_x=v_0\cos{\theta}\) and \(v_y=v_0\sin{\theta}\).
05

Equate the ratios of the components

Now, we set up the proportionality condition as follows: \(\frac{v_x}{a_x} = \frac{v_y}{a_y}\), which leads to \(\frac{v_0\cos{\theta}}{0} = \frac{v_0\sin{\theta}}{-g}\).
06

Analyze the proportionality condition

Since \(v_0\) is non-zero and \(g\) is also non-zero, we can ignore them in the equation and analyze the ratio of the trigonometric functions \(\cos{\theta}\) and \(\sin{\theta}\). However, since the denominator \(a_x=0\), the equation leads to an undefined expression, which means that no angle \(\theta\) between \(0^{\circ}\) and \(90^{\circ}\) can fulfill the condition for parallel vectors in this case. So, the answer is: e) none of the angles

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Most popular questions from this chapter

A circus juggler performs an act with balls that he tosses with his right hand and catches with his left hand. Each ball is launched at an angle of \(75^{\circ}\) and reaches a maximum height of \(90 \mathrm{~cm}\) above the launching height. If it takes the juggler \(0.2 \mathrm{~s}\) to catch a ball with his left hand, pass it to his right hand and toss it back into the air, what is the maximum number of balls he can juggle?

To attain maximum height for the trajectory of a projectile, what angle would you choose between \(0^{\circ}\) and \(90^{\circ}\), assuming that you can launch the projectile with the same initial speed independent of the launch angle. Explain your reasoning.

A particle's motion is described by the following two parametric equations: $$ \begin{array}{l} x(t)=5 \cos (2 \pi t) \\ y(t)=5 \sin (2 \pi t) \end{array} $$ where the displacements are in meters and \(t\) is the time, in seconds. a) Draw a graph of the particle's trajectory (that is, a graph of \(y\) versus \(x\) ). b) Determine the equations that describe the \(x\) - and \(y\) -components of the velocity, \(v_{x}\) and \(v_{y}\), as functions of time. c) Draw a graph of the particle's speed as a function of time.

A cruise ship moves southward in still water at a speed of \(20.0 \mathrm{~km} / \mathrm{h},\) while a passenger on the deck of the ship walks toward the east at a speed of \(5.0 \mathrm{~km} / \mathrm{h}\). The passenger's velocity with respect to Earth is a) \(20.6 \mathrm{~km} / \mathrm{h}\), at an angle of \(14.04^{\circ}\) east of south. b) \(20.6 \mathrm{~km} / \mathrm{h}\), at an angle of \(14.04^{\circ}\) south of east. c) \(25.0 \mathrm{~km} / \mathrm{h}\), south. d) \(25.0 \mathrm{~km} / \mathrm{h}\), east. e) \(20.6 \mathrm{~km} / \mathrm{h}\), south.

Salmon often jump upstream through waterfalls to reach their breeding grounds. One salmon came across a waterfall \(1.05 \mathrm{~m}\) in height, which she jumped in \(2.1 \mathrm{~s}\) at an angle of \(35^{\circ}\) to continue upstream. What was the initial speed of her jump?
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