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Chapter 2: Problem 99

A particle can be projected with a given speed in two possible ways so as to make it pass through a point at a distance \(r\) from the point of projection. The product of the times taken to reach this point in the two possible ways is then proportional to (A) \(r\) (B) \(\frac{1}{r}\) (C) \(\frac{1}{r^{2}}\) (D) \(\frac{1}{r^{3}}\)

Short Answer

Expert verified
The product of times t1 and t2 is proportional to \(r^2 \). However, none of the given options match this result, which suggests an error in the options. The correct option should be (E) \(r^2\).

Step by step solution

01

Understanding the projectile motion

A projectile motion can be described as a motion of an object under the influence of gravity. In this problem, the particle can be projected with a given speed to pass through a point at a distance r from the point of projection. From the point of projection, there are two possible ways: one at an angle 胃, and the other at an angle (90掳 - 胃). Let's consider the given speed as v, the initial velocity components are v sin(胃) and v cos(胃). We can denote the time taken for the two possible ways as t1 and t2.
02

Calculate time taken to reach point

In the horizontal plane, the particle will travel a distance r. We can write the equation for the horizontal distance covered by the particle as: For the first possible way: r = v cos(胃) * t1 For the second possible way: r = v sin(胃) * t2
03

Calculate the product of times t1 and t2

We need to calculate the product of t1 and t2. From the equations in Step 2, we get: t1 = r / (v cos(胃)) t2 = r / (v sin(胃)) Now, calculate the product t1 * t2: t1 * t2 = (r / (v cos(胃))) * (r / (v sin(胃))) t1 * t2 = r^2 / (v^2 sin(胃) cos(胃))
04

Analyze the proportionality

The product of times t1 and t2 is proportional to: t1 * t2 鈭� r^2 / (sin(胃) cos(胃)) Since v is constant, and (sin(胃) cos(胃)) depends on the angle, not the distance r, we can conclude that the product of times t1 and t2 is proportional to r^2.
05

Compare with the given options

We have found that the product of times t1 and t2 is proportional to r^2. Let's compare this with the given options: (A) r (B) 1/r (C) 1/r^2 (D) 1/r^3 Clearly, none of the options match the obtained proportionality. There might be an error in the given options. The correct option should be: (E) r^2

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Most popular questions from this chapter

A particle starts from the origin at \(t=0 \mathrm{~s}\) with a velocity of \(10.0 \hat{j} \mathrm{~m} / \mathrm{s}\) and moves in the \(x-y\) plane with a constant acceleration of \((8.0 \hat{i}+2.0 \hat{j}) \mathrm{ms}^{-2}\) At a time when \(x\)-coordinate of the particle \(16 \mathrm{~m}\), (A) \(24 \mathrm{~m}\) (B) \(22 \mathrm{~m}\) (C) \(20 \mathrm{~m}\) (D) None

A particle is projected upwards with a velocity of \(100 \mathrm{~m} / \mathrm{s}\) at an angle of \(37^{\circ}\) with the vertical. The time when the particle will move perpendicular to its initial direction is \(\left(g=10 \mathrm{~m} / \mathrm{s}^{2}, \tan 53^{\circ}=4 / 3\right)\) (A) \(10 \mathrm{~s}\) (B) \(12.5 \mathrm{~s}\) (C) \(15 \mathrm{~s}\) (D) \(16 \mathrm{~s}\)

A boat, which has a speed of \(5 \mathrm{~km} / \mathrm{h}\) in still water, crosses a river of width \(1 \mathrm{~km}\) along the shortest possible path in 15 minutes. The velocity of the river water in kilometers per hour is (A) 1 (B) 3 (C) 4 (D) \(\sqrt{41}\)

A particle is projected at an angle \(\alpha\) with the horizontal from the foot of an inclined plane making an angle \(\beta\) with horizontal. Which of the following expressions holds good if the particle strikes the inclined plane normally? (A) \(\cot \beta=\tan (\alpha-\beta)\) (B) \(\cot \beta=2 \tan (\alpha-\beta)\) (C) \(\cot \alpha=\tan (\alpha-\beta)\) (D) \(\cot \alpha=2 \tan (\alpha-\beta)\)

A projectile can have the same range \(R\) for two angles of projection. If \(t_{1}\) and \(t_{2}\) be the times of flights in the two cases, then the product of the two times of flights is proportional to \([\mathbf{2 0 0 5}]\) (A) \(\frac{1}{R^{2}}\) (B) \(R^{2}\) (C) \(R\) (D) \(\frac{1}{R}\)
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