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Chapter 3: Problem 94

A plane diving with constant speed at an angle of \(49.0^{\circ}\) with the vertical, releases a package at an altitude of \(600 . \mathrm{m}\). The package hits the ground \(3.50 \mathrm{~s}\) after release. How far horizontally does the package travel?

Short Answer

Expert verified
Answer: The package travels approximately 359 meters horizontally before hitting the ground.

Step by step solution

01

Determine the angle and altitude in radians and meters respectively

First, convert the angle from degrees to radians since it is more suitable for calculations involving angles. Angle in radians = \((49.0^{\circ}) \times (\pi/180) = 0.855 \mathrm{rad}\) Now let's consider the altitude of the package. Altitude = \(600.0 \mathrm{m}\)
02

Determine the vertical velocity component

From the given angle, we can determine the vertical velocity component: \(V_v = V \times \sin(0.855)\) We need to find the value of V to calculate the vertical velocity component. To do this, we can use the equation for the time it takes for the package to hit the ground: \(t=3.50 \mathrm{s}\) For vertical motion, we know that \(s = ut + \frac{1}{2}at^2\) Using the vertical motion equation: \(600 = V_v \times 3.50 - \frac{1}{2} \times 9.81 \times (3.50)^2\) We can now solve for \(V_v\). \(V_v = \frac{600 + \frac{1}{2} \times 9.81 \times (3.50)^2}{3.50} = 183.495 \mathrm{m/s}\)
03

Determine the total velocity V

Now we can use the value of \(V_v\) to find the total velocity \(V\) as follows: \(V_v = V \times \sin(0.855)\) \(V = \frac{V_v}{\sin(0.855)} = \frac{183.495}{\sin(0.855)} = 202.152 \mathrm{m/s}\)
04

Determine the horizontal velocity component

With the total velocity \(V\), we can now calculate the horizontal velocity component: \(V_h = V \times \cos(0.855) = 202.152 \times \cos(0.855) = 102.563 \mathrm{m/s}\)
05

Calculate the horizontal distance

Finally, we can use the horizontal velocity component and the time it takes for the package to hit the ground to find the horizontal distance traveled: Horizontal Distance = \(V_h \times t = 102.563 \times 3.50 = 358.971 \mathrm{m}\) So the package travels approximately \(359 \mathrm{m}\) horizontally before hitting the ground.

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Most popular questions from this chapter

In ideal projectile motion, when the positive \(y\) -axis is chosen to be vertically upward, the \(y\) -component of the velocity of the object during the ascending part of the motion and the \(y\) -component of the velocity during the descending part of the motion are, respectively, a) positive, negative. c) positive, positive. b) negative, positive. d) negative, negative.

You want to cross a straight section of a river that has a uniform current of \(5.33 \mathrm{~m} / \mathrm{s}\) and is \(127, \mathrm{~m}\) wide. Your motorboat has an engine that can generate a speed of \(17.5 \mathrm{~m} / \mathrm{s}\) for your boat. Assume that you reach top speed right away (that is, neglect the time it takes to accelerate the boat to top speed). a) If you want to go directly across the river with a \(90^{\circ}\) angle relative to the riverbank, at what angle relative to the riverbank should you point your boat? b) How long will it take to cross the river in this way? c) In which direction should you aim your boat to achieve minimum crossing time? d) What is the minimum time to cross the river? e) What is the minimum speed of your boat that will still enable you to cross the river with a \(90^{\circ}\) angle relative to the riverbank?

During a jaunt on your sailboat, you sail \(2.00 \mathrm{~km}\) east, \(4.00 \mathrm{~km}\) southeast, and an additional distance in an unknown direction. Your final position is \(6.00 \mathrm{~km}\) directly east of the starting point. Find the magnitude and direction of the third leg of your journey.

Your friend's car is parked on a cliff overlooking the ocean on an incline that makes an angle of \(17.0^{\circ}\) below the horizontal. The brakes fail, and the car rolls from rest down the incline for a distance of \(29.0 \mathrm{~m}\) to the edge of the cliff, which is \(55.0 \mathrm{~m}\) above the ocean, and, unfortunately, continues over the edge and lands in the ocean. a) Find the car's position relative to the base of the cliff when the car lands in the ocean. b) Find the length of time the car is in the air.

For a given initial speed of an ideal projectile, there is (are) \(\quad\) launch angle(s) for which the range of the projectile is the same. a) only one b) two different c) more than two but a finite number of d) only one if the angle is \(45^{\circ}\) but otherwise two different e) an infinite number of
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