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Chapter 3: Problem 62

A bullet is fired from a rifle that is held \(1.6 \mathrm{~m}\) above the ground in a horizontal position. The initial speed of the bullet is \(1100 \mathrm{~m} / \mathrm{s}\). Find (a) the time it takes for the bullet to strike the ground and (b) the horizontal distance traveled by the bullet.

Short Answer

Expert verified
(a) Time to strike the ground: 0.571 s; (b) Horizontal distance: 628.1 m.

Step by step solution

01

Determine Time to Strike the Ground

To find the time it takes for the bullet to strike the ground, we need to use the formula for the time of free fall: \[ t = \sqrt{\frac{2h}{g}} \]where \( h = 1.6 \, \text{m} \) is the height and \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity.Plug in the given values:\[ t = \sqrt{\frac{2 \times 1.6}{9.8}} \approx \sqrt{0.3265} \approx 0.571 \, \text{s}. \]
02

Calculate Horizontal Distance

The horizontal distance traveled by the bullet can be found using the formula:\[ d = v \times t \]where\( v = 1100 \, \text{m/s} \) is the initial speed of the bullet.Using the time from Step 1,\[ d = 1100 \times 0.571 \approx 628.1 \, \text{m}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Free Fall
When we talk about free fall, we're discussing an object's movement under the influence of gravity alone. Here, gravity accelerates the object downwards without any initial vertical velocity. In our exercise, the bullet is fired straight and horizontally. Its vertical motion is solely influenced by gravity, hence it undergoes free fall.
  • Gravity: The constant force pulling objects towards the Earth. This is often denoted by "g" and is approximately equal to \(9.8 \, \text{m/s}^2\).
  • Free Fall Equation: To find the time it takes for an object to hit the ground, the equation \( t = \sqrt{\frac{2h}{g}} \) is used. Here, "h" represents the height from which the object is dropped.
In our exercise, free fall dictates how quickly the bullet descends from its initial height. To find the time taken for the bullet to strike the ground, we plug in the height of \(1.6 \, \text{m}\) and solve the equation for "t". The result shows that it takes about \(0.571 \, \text{s}\) for the bullet to hit the ground.
Horizontal Distance
The horizontal distance refers to how far an object travels along the horizontal axis. In projectile motion, even as objects fall, they may also travel horizontally, which is dominated by the initial horizontal velocity since gravity does not affect horizontal speeds directly.
  • Initial Speed: The speed at which the bullet is fired plays a crucial role. In this exercise, it's given as \(1100 \, \text{m/s}\).
  • Horizontal Distance Formula: The formula \( d = v \times t \) helps calculate the overall horizontal distance covered by a projectile over a given time "t".
In this context, as the bullet travels forward at an initial speed of \(1100 \, \text{m/s}\), it simultaneously falls. Using the time of flight from its free fall calculation, we can determine that it covers approximately \(628.1 \, \text{m}\) horizontally before hitting the ground.
Time of Flight
The time of flight is simply the duration for which the projectile stays in motion before hitting the ground. For horizontally fired projectiles, the time of flight depends entirely on the vertical motion of the object.
  • Role in Projectile Motion: It links the vertical drop to horizontal motion.
  • Calculation: In our bullet's scenario, time of flight is derived using the free fall equation as height and gravity dictate this duration.
Understanding the time of flight enables us to connect how far and how long the bullet will travel before it strikes the ground, assisting in calculating the horizontal distance given an unaltered horizontal speed throughout its motion.

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Most popular questions from this chapter

A major-league pitcher can throw a baseball in excess of \(41.0 \mathrm{~m} / \mathrm{s}\). If a ball is thrown horizontally at this speed, how much will it drop by the time it reaches a catcher who is \(17.0 \mathrm{~m}\) away from the point of release?

A rifle is used to shoot twice at a target, using identical cartridges. The first time, the rifle is aimed parallel to the ground and directly at the center of the bull's-eye. The bullet strikes the target at a distance of \(H_{\mathrm{A}}\) below the center, however. The second time, the rifle is similarly aimed, but from twice the distance from the target. This time the bullet strikes the target at a distance of \(H_{\mathrm{B}}\) below the center. Find the ratio \(H_{\mathrm{B}} / H_{\mathrm{A}}\).

(a) Does Barbara see him moving toward the east or toward the west? (b) Does Barbara see him moving toward the north or toward the south? (c) Considering your answers to parts (a) and (b), how does Barbara see Neil moving relative to herself, toward the east and north, toward the east and south, toward the west and north, or toward the west and south? Justify your answers in each case. With respect to the ground, Barbara is skating due south at a speed of \(4.0 \mathrm{~m} / \mathrm{s}\). With respect to the ground, Neil is skating due west at a speed of \(3.2 \mathrm{~m} / \mathrm{s} .\) Find Neil's velocity (magnitude and direction relative to due west) as seen by Barbara. Make sure that your answer agrees with your answer to part (c) of the Concept Questions.

A puck is moving on an air hockey table. Relative to an \(x, y\) coordinate system at time \(t=0 \mathrm{~s},\) the \(x\) components of the puck's initial velocity and acceleration are \(v_{0 x}=+1.0 \mathrm{~m} / \mathrm{s}\) and \(a_{x}=+2.0 \mathrm{~m} / \mathrm{s}^{2} .\) The \(y\) components of the puck's initial velocity and acceleration are \(v_{0 y}=+2.0 \mathrm{~m} / \mathrm{s}\) and \(a_{y}=-2.0 \mathrm{~m} / \mathrm{s}^{2}\) Is the magnitude of the \(x\) component of the velocity increasing or decreasing in time? Is the magnitude of the \(y\) component of the velocity increasing or decreasing in time? Find the magnitude and direction of the puck's velocity at a time of \(t=0.50 \mathrm{~s}\). Specify the direction relative to the \(+x\) axis. Be sure that your calculations are consistent with your answers to the Concept Questions.

An Olympic long jumper leaves the ground at an angle of \(23^{\circ}\) and travels through the air for a horizontal distance of \(8.7 \mathrm{~m}\) before landing. What is the takeoff speed of the jumper?
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