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Magazine Chapter 3: Problem 31
An eagle is flying horizontally at \(6.0 \mathrm{m} / \mathrm{s}\) with a fish in its claws. It accidentally drops the fish. (a) How much time passes before the fish's speed doubles? (b) How much additional time would be required for the fish's speed to double again?
Short Answer
Expert verified
(a) 1.06 s; (b) 1.31 s
Step by step solution
01
Understand the Problem
The eagle is flying horizontally with a velocity of \(6.0\,\mathrm{m/s}\). The fish is dropped, so it will start accelerating downward due to gravity. We need to find out the time taken for the fish's speed to double and then double again.
02
Determine Initial Conditions
Initially, the velocity of the fish is horizontal, \(v_x = 6.0\,\mathrm{m/s}\), and there is no initial vertical velocity, \(v_y = 0\). After being dropped, the fish accelerates downward due to gravity with an acceleration of \(g = 9.8\,\mathrm{m/s}^2\).
03
Write the Equation for Vertical Velocity
The vertical velocity of the fish at time \(t\) is given by the equation: \[ v_y = g \cdot t \]Substituting \(g = 9.8\,\mathrm{m/s}^2\):\[ v_y = 9.8 \cdot t \]
04
Calculate the Total Speed at Time t
The total speed of the fish at any time \(t\) is given by the Pythagorean theorem: \[ v = \sqrt{{v_x}^2 + {v_y}^2} \]Substituting the known values:\[ v = \sqrt{(6.0)^2 + (9.8t)^2} \]
05
Solve for Time when Speed Doubles
Initially, the speed is \(6.0\,\mathrm{m/s}\). The speed doubles to \(12.0\,\mathrm{m/s}\). We solve this equation:\[ \sqrt{(6.0)^2 + (9.8t)^2} = 12.0 \]Squaring both sides:\[ 36 + 96.04 t^2 = 144 \]Rearranging:\[ 96.04 t^2 = 144 - 36 \]\[ 96.04 t^2 = 108 \]\[ t^2 = \frac{108}{96.04} \]\[ t = \sqrt{1.125} \]\[ t \approx 1.06\, \mathrm{s} \]
06
Solve for Time when Speed Doubles Again
Next, we find the time for the speed to double again to \(24.0\,\mathrm{m/s}\):\[ \sqrt{(6.0)^2 + (9.8t)^2} = 24.0 \]Squaring both sides:\[ 36 + 96.04 t^2 = 576 \]Rearranging:\[ 96.04 t^2 = 576 - 36 \]\[ 96.04 t^2 = 540 \]\[ t^2 = \frac{540}{96.04} \]\[ t = \sqrt{5.625} \]\[ t \approx 2.37\,\mathrm{s} \]
07
Calculate Additional Time Required
To find the additional time required for the speed to double a second time:\[ \text{Additional time} = 2.37\,\mathrm{s} - 1.06\,\mathrm{s} = 1.31\,\mathrm{s} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Free Fall
When the eagle accidentally drops the fish from its claws, the fish experiences a physical phenomenon known as free fall.
In free fall, an object moves only under the influence of gravity, with no initial vertical velocity, because the fish starts from rest vertically.
Free fall can occur in any direction but it always accelerates downwards towards the Earth. - Gravity is the only force acting on the fish, ignoring air resistance for simplicity. - The fish starts with an initial velocity of zero in the vertical direction since being held horizontally. - Once in free fall, the fish鈥檚 motion can be analyzed using basic kinematics equations ideal for uniform acceleration or constant gravity. Free fall is important as it introduces the concept that objects, regardless of mass, fall with the same acceleration in the absence of air resistance. This fundamental observation was famously illustrated by Galileo in his work.
In free fall, an object moves only under the influence of gravity, with no initial vertical velocity, because the fish starts from rest vertically.
Free fall can occur in any direction but it always accelerates downwards towards the Earth. - Gravity is the only force acting on the fish, ignoring air resistance for simplicity. - The fish starts with an initial velocity of zero in the vertical direction since being held horizontally. - Once in free fall, the fish鈥檚 motion can be analyzed using basic kinematics equations ideal for uniform acceleration or constant gravity. Free fall is important as it introduces the concept that objects, regardless of mass, fall with the same acceleration in the absence of air resistance. This fundamental observation was famously illustrated by Galileo in his work.
Velocity
Velocity in projectile motion consists of two components: horizontal and vertical. The horizontal component initially depends on the velocity imparted by the eagle, which is constant if ignoring air resistance. Meanwhile, the vertical component changes over time because gravity acts on it.
- **Horizontal Velocity**
- For the eagle, this is a constant 6.0 m/s.
- Since there is no air resistance factored, it doesn鈥檛 change throughout the motion.
- **Vertical Velocity**
- Initially zero as the fish starts its descent.
- Increases at a rate determined by the acceleration due to gravity.
To find the fish's overall velocity at any given time, we combine these two components using the Pythagorean theorem.
Understanding velocity is crucial as it combines both speed and direction, giving a complete description of an object's motion.
Acceleration due to Gravity
Acceleration due to gravity is a constant acceleration experienced by objects in free fall near the Earth's surface. The value of this acceleration is approximately 9.8 m/s虏, pointing downwards.- Acts equally on all objects regardless of mass.- Responsible for the increasing vertical velocity of the fish.This acceleration influences how the velocity of the fish changes with time:- **Equation of Motion**: Vertical velocity (\(v_y\)) at any time (\(t\)) is calculated as: \[ v_y = g \cdot t \] \(g\) is the acceleration due to gravity.- Understanding this constant acceleration factor helps predict how quickly an object will accelerate as it falls, and is a key component in calculating the time needed for the velocity to reach certain thresholds in problems like this one.
