Chapter 4: Problem 86
A crow is flying horizontally with a constant speed of
Short Answer
Expert verified
Horizontal velocity: 2.70 m/s, Vertical velocity: 20.601 m/s.
Step by step solution
01
Understand the Problem
A crow is flying with a constant horizontal speed and releases a clam which follows projectile motion. We need to find the horizontal and vertical components of the clam's velocity just before it hits the ground.
02
Identify Initial Horizontal Velocity
Since the clam is released from the crow's beak, it inherits the horizontal velocity of the crow. Thus, the initial horizontal velocity, , is .
03
Constant Horizontal Velocity
In projectile motion, there is no horizontal acceleration (assuming air resistance is negligible), so the horizontal component of velocity remains constant throughout the motion. Thus, .
04
Calculate Time of Flight
The time of flight is given as , which is the time the clam takes from being released to landing.
05
Determine Vertical Component of Velocity
The initial vertical velocity, , is since the clam is not initially moving up or down. Using the equation where (acceleration due to gravity) and , we find the vertical velocity just before landing: .
06
Summarize Velocity Components
The horizontal velocity component just before landing is and the vertical velocity component is .
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Horizontal Velocity
When studying projectile motion, one of the important aspects is understanding the behavior of horizontal velocity. In the case of the crow releasing a clam while flying at a constant speed, the horizontal velocity is particularly straightforward. Since the crow is moving at a steady pace of as it releases the clam, this velocity becomes the initial horizontal velocity of the clam.
In the absence of air resistance, which simplifies calculations in many physics problems, the horizontal velocity stays constant throughout the clam’s journey.
To break it down, here's why the horizontal velocity remains unchanged: . Understanding this concept illustrates a key principle in physics: without an external force in the horizontal direction, velocity remains constant.
In the absence of air resistance, which simplifies calculations in many physics problems, the horizontal velocity stays constant throughout the clam’s journey.
To break it down, here's why the horizontal velocity remains unchanged:
- There is no horizontal force acting on the clam after it leaves the crow's beak.
- In projectile motion, horizontal acceleration is zero.
Vertical Velocity
Vertical velocity in projectile motion is more dynamic compared to horizontal velocity because it is directly influenced by gravity.
When the clam is initially released by the crow, it has no vertical velocity, or in other words, it isn't moving up or down yet, so the initial vertical velocity is .
However, once the clam is in free fall, gravity begins to accelerate it downward at .
To calculate the vertical velocity of the clam just before impact, we employ the equation for motion under constant acceleration: where: Thus, the clam’s vertical velocity just before it hits the beach is , moving downward due to the influence of gravity.
When the clam is initially released by the crow, it has no vertical velocity, or in other words, it isn't moving up or down yet, so the initial vertical velocity is
However, once the clam is in free fall, gravity begins to accelerate it downward at
To calculate the vertical velocity of the clam just before impact, we employ the equation for motion under constant acceleration:
is the initial vertical velocity ( for the clam). is the acceleration due to gravity . is the time of flight, .
Acceleration Due to Gravity
Gravity is a key player in the story of projectile motion. It's this mysterious force that gives weight to objects and is responsible for the complex trajectory paths we see in projectiles like our clam.
Gravity exerts a constant acceleration on any object in free fall towards Earth, quantified as . This value tells us how much faster an object is moving vertically with each passing second.
In projectile motion, gravity specifically impacts the vertical component of velocity, causing even a horizontally launched object to eventually accelerate towards ground.
Gravity exerts a constant acceleration on any object in free fall towards Earth, quantified as
In projectile motion, gravity specifically impacts the vertical component of velocity, causing even a horizontally launched object to eventually accelerate towards ground.
- This acceleration doesn't affect horizontal velocity; it only changes the vertical. That's why the horizontal velocity remains steady as we’ve discussed.
- Starting with no vertical motion, our clam is pulled down by gravity, gaining vertical speed every second it is airborne.