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Chapter 4: Problem 18

A white-crowned sparrow flying horizontally with a speed of \(1.80 \mathrm{m} / \mathrm{s}\) folds its wings and begins to drop in free fall. (a) How far does the sparrow fall after traveling a horizontal distance of \(0.500 \mathrm{m} ?\) (b) If the sparrow's initial speed is increased, does the distance of fall increase, decrease, or stay the same?

Short Answer

Expert verified
(a) Approximately 0.378 m. (b) Distance of fall stays the same.

Step by step solution

01

Determine Time to Travel Horizontal Distance

First, we calculate the time it takes for the sparrow to travel the horizontal distance of 0.500 m with a speed of 1.80 m/s. Using the formula: \[ t = \frac{\text{distance}}{\text{speed}} = \frac{0.500 \text{ m}}{1.80 \text{ m/s}} \]Calculate this to find the time.
02

Calculate Time Duration

Perform the calculation:\[ t = \frac{0.500}{1.80} \approx 0.2778 \text{ seconds} \]So, the time taken to cover 0.500 m horizontally is approximately 0.2778 seconds.
03

Apply Free Fall Formula

Use the formula for distance under free fall, which is:\[ d = \frac{1}{2} g t^2 \]where \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity and \( t = 0.2778 \text{ seconds} \) is the time we calculated.
04

Calculate Vertical Distance Fallen

Substitute the values into the free fall formula:\[ d = \frac{1}{2} \times 9.81 \times (0.2778)^2 \]Calculate this to find the distance fallen.
05

Solve for Distance

Perform the calculation:\[ d = \frac{1}{2} \times 9.81 \times 0.0772 \approx 0.378 \text{ meters} \]So, the sparrow falls approximately 0.378 meters.
06

Analyze Effect of Speed on Distance of Fall

For part (b), since the vertical fall depends only on the time in free fall determined by horizontal distance traveled and speed, changing the initial horizontal speed affects the time taken to cover the horizontal distance but not the vertical distance fallen within the same horizontal distance. Thus, the distance of fall stays the same regardless of the initial horizontal speed.

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Key Concepts

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

Free Fall
When an object is in free fall, it is only influenced by gravity, with no air resistance affecting its motion. Let's imagine our sparrow deciding to stop flapping its wings; this means it stops exerting any additional force to stay aloft and starts to be pulled down exclusively by gravity. At this point, it transitions into free fall. In a vacuum, all objects in free fall accelerate downward at the same rate, regardless of their mass. Here on Earth, this acceleration is due to the force of gravity.
Horizontal Velocity
Horizontal velocity refers to the constant speed at which an object moves parallel to the horizontal plane. In our scenario, the sparrow initially flies at a constant horizontal velocity of 1.80 m/s. Horizontal velocity can be considered separately from vertical motion when discussing projectile motion because gravity only affects the vertical component. Thus, no matter how fast the sparrow is flying horizontally, its horizontal velocity remains unchanged unless acted upon by an external force.
Acceleration Due to Gravity
The acceleration due to gravity is the rate at which an object increases its velocity as it falls towards the Earth. This constant acceleration on Earth is approximately 9.81 m/s虏. When you release an object, like our sparrow in free fall, gravity accelerates it downward. This acceleration is what dictates how quickly the sparrow falls even though it continues to have horizontal velocity. Therefore, acceleration due to gravity is a crucial factor in calculating how far something will fall in a given timeframe of free fall.
Distance Calculation
Calculating the distance an object falls while in free fall involves understanding the equation of motion. The primary formula used is:
  • \[ d = \frac{1}{2} g t^2 \]
Here, "d" is the distance fallen, "g" is the acceleration due to gravity, and "t" is the time the object has been in free fall. In our exercise, the sparrow falls under the influence of gravity for about 0.2778 seconds as it travels a horizontal distance of 0.500 meters. By plugging these values into our formula, we find the sparrow falls approximately 0.378 meters. This calculation confirms that the fall distance doesn't change with horizontal speed since the fall time determined its vertical drop.

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Most popular questions from this chapter

On a hot summer day, a young girl swings on a rope above the local swimming hole (Figure \(4-20\) ). When she lets go of the rope her initial velocity is \(2.25 \mathrm{m} / \mathrm{s}\) at an angle of \(35.0^{\circ}\) above the horizontal. If she is in flight for \(0.616 \mathrm{s}\), how high above the water was she when she let go of the rope?

A hot-air balloon rises from the ground with a velocity of\((2.00 \mathrm{m} / \mathrm{s}) \hat{\mathrm{y}}\) A champagne bottle is opened to celebrate takeoff, expelling the cork horizontally with a velocity of \((5.00 \mathrm{m} / \mathrm{s}) \hat{\mathrm{x}}\) relative to the balloon. When opened, the bottle is \(6.00 \mathrm{m}\) above the ground. (a) What is the initial velocity of the cork, as seen by an observer on the ground? Give your answer in terms of the \(x\) and \(y\) unit vectors. (b) What are the speed of the cork and its initial direction of motion as seen by the same observer? (c) Determine the maximum height above the ground attained by the cork. (d) How long does the cork remain in the air?

Pitcher's mounds are raised to compensate for the vertical drop of the ball as it travels a horizontal distance of \(18 \mathrm{m}\) to the catcher. (a) If a pitch is thrown horizontally with an initial speed of \(32 \mathrm{m} / \mathrm{s}\), how far does it drop by the time it reaches the catcher? (b) If the speed of the pitch is increased, does the drop distance increase, decrease, or stay the same? Explain. (c) If this baseball game were to be played on the Moon, would the drop distance increase, decrease, or stay the same? Explain.

Astronomers have discovered several volcanoes on Io, a moon of Jupiter. One of them, named Loki, ejects lava to a maximum height of \(2.00 \times 10^{5} \mathrm{m}\). (a) What is the initial speed of the lava? (The acceleration of gravity on lo is \(1.80 \mathrm{m} / \mathrm{s}^{2}\).) (b) If this volcano were on Earth, would the maximum height of the ejected lava be greater than, less than, or the same as on Io? Explain.

A crow is flying horizontally with a constant speed of \(2.70 \mathrm{m} / \mathrm{s}\) when it releases a clam from its beak (Figure \(4-14) .\) The clam lands on the rocky beach \(2.10 \mathrm{s}\) later. Just before the clam lands, what is (a) its horizontal component of velocity, and (b) its vertical component of velocity? (c) How would your answers to parts (a) and (b) change if the speed of the crow were increased? Explain.
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