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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

The air speed indicator of a plane that took off from Detroit reads \(350 . \mathrm{km} / \mathrm{h}\) and the compass indicates that it is heading due east to Boston. A steady wind is blowing due north at \(40.0 \mathrm{~km} / \mathrm{h}\). Calculate the velocity of the plane with reference to the ground. If the pilot wishes to fly directly to Boston (due east) what must the compass read?

A projectile is launched at a \(60^{\circ}\) angle above the horizontal on level ground. The change in its velocity between launch and just before landing is found to be \(\Delta \vec{v}=\vec{v}_{\text {landing }}-\vec{v}_{\text {launch }}=-20 \hat{y} \mathrm{~m} / \mathrm{s}\). What is the initial velocity of the projectile? What is its final velocity just before landing?

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