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Chapter 4: Problem 16

You throw a ball from a cliff with an initial velocity of \(15 \mathrm{~m} / \mathrm{s}\) at an angle of \(20^{\circ}\) below the horizontal. Find \((a)\) its horizontal displacement and \((b)\) its vertical displacement \(2.3\) s later

Short Answer

Expert verified
The horizontal displacement of the ball after 2.3 seconds is approximately 33.02 meters, and the vertical displacement is approximately -16.32 meters.

Step by step solution

01

Find the Horizontal and Vertical Component of Initial Velocity

First, remember that the initial velocity has both a horizontal and vertical component. For an angle below the horizontal, the horizontal component \(v_{ix}\) can be calculated as \(v_{i} * cos(\theta)\) where \(v_{i}=15 \mathrm{m/s}\) is the magnitude of initial velocity and \(\theta = 20^{\circ}\). Similarly, the vertical component \(v_{iy}\) can be calculated as \(v_{i} * sin(\theta)\). Be mindful that because the angle is below the horizontal the vertical component will be negative, as it directs downwards.
02

Calculate Horizontal Displacement

In projectile motion, horizontal displacement is given by \(x = v_{ix} * t\), where \(x\) is the horizontal displacement, \(v_{ix}\) is the horizontal component of initial velocity and \(t=2.3s\) is the time. Substitute the values into the equation to get the horizontal displacement.
03

Calculate Vertical Displacement

In projectile motion, vertical displacement is given by \(y = v_{iy} * t + 0.5 * g * t^{2}\), where \(y\) is the vertical displacement, \(v_{iy}\) is vertical component of initial velocity, \(t=2.3s\) is the time and \(g=9.8 \mathrm{m/s^2}\) is the acceleration due to gravity. Substitute the values into the equation to get the vertical displacement. Keep in mind that the displacement is negative as the motion is downward.

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Most popular questions from this chapter

A train travels due south at \(28 \mathrm{~m} / \mathrm{s}\) (relative to the ground) in rain that is blown to the south by the wind. The path of each raindrop makes an angle of \(64^{\circ}\) with the vertical, as measured by an observer stationary on the Earth. An observer on the train, however, sees perfectly vertical tracks of rain on the windowpane. Determine the speed of the drops relative to the Earth.

An elevator ascends with an upward acceleration of \(4.0-\mathrm{ft} / \mathrm{s}^{2}\). At the instant its upward speed is \(8.0 \mathrm{ft} / \mathrm{s}\), a loose bolt drops from the ceiling of the elevator \(9.0 \mathrm{ft}\) from the floor. Calculate (a) the time of flight of the bolt from ceiling to floor and \((b)\) the distance it has fallen relative to the elevator shaft.

A transcontinental flight at \(2700 \mathrm{mi}\) is scheduled to take 50 min longer westward than eastward. The air speed of the jet is \(600 \mathrm{mi} / \mathrm{h}\). What assumptions about the jet-stream wind velocity, presumed to be east or west, are made in preparing the schedule?

In a baseball game, a batter hits the ball at a height of \(4.60 \mathrm{ft}\) above the ground so that its angle of projection is \(52.0^{\circ}\) to the horizontal. The ball lands in the grandstand, \(39.0 \mathrm{ft}\) up from the bottom; see Fig. 4-38. The grandstand seats slope upward at \(28.0^{\circ}\) with the bottom seats \(358 \mathrm{ft}\) from home plate. Calculate the speed with which the ball left the bat. (Ignore air resistance.)

If the pitcher's mound is \(1.25 \mathrm{ft}\) above the baseball field, can a pitcher release a fast ball horizontally at \(92.0 \mathrm{mi} / \mathrm{h}\) and still get it into the strike zone over the plate \(60.5 \mathrm{ft}\) away? Assume that, for a strike, the ball must fall at least \(1.30 \mathrm{ft}\) but no more than \(3.60 \mathrm{ft}\).
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