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Chapter 2: Problem 7

An object is dropped from altitude \(y_{0}\). Determine the impact velocity if air resistance is neglected-that is, if we assume no drag force.

Short Answer

Expert verified
Answer: The impact velocity of the object when it reaches the ground is given by the expression \(v = \sqrt{19.62y_{0}}\).

Step by step solution

01

Understand the given information

We are given that an object is dropped from an initial altitude (\(y_{0}\)) and there is no air resistance. The only force acting on the object is gravity, which will cause an acceleration of \(g\) (\(9.81 m/s^2\)).
02

Choose the kinematic equation to solve for final velocity

As we are dealing with constant acceleration due to gravity (\(g\)), we can use kinematic equations. The equation that relates initial position, initial velocity, acceleration, final position, and final velocity is: \(v^2 = u^2 + 2as\), where: - \(v\) is the final velocity - \(u\) is the initial velocity - \(a\) is the acceleration (\(g\) in this case, downward) - \(s\) is the change in position (altitude)
03

Plug in the given values and solve for final velocity

As the object is dropped, its initial velocity (\(u\)) is \(0 m/s\). The acceleration (\(a\)) is \(g\) (\(9.81 m/s^2\)), and the altitude change (\(s\)) is equal to -\(y_{0}\) (negative since it is downward motion). Now let's plug in the given values into our equation: \(v^2 = u^2 + 2as\) \(v^2 = 0^2 + 2(-9.81)(-y_{0})\) \(v^2 = 19.62y_{0}\) Now we just need to take the square root of both sides to find the impact velocity (\(v\)): \(v = \pm\sqrt{19.62y_{0}}\) The negative root corresponds to initial drop and the positive root to impact (since the direction is downward). So, the impact velocity is: \(v = \sqrt{19.62y_{0}}\) Now we have found an expression for the impact velocity of the object when it reaches the ground.

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Key Concepts

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

Kinematic Equations
When objects move with a constant acceleration, we can describe their motion using kinematic equations. These equations are the core tools in classical mechanics for predicting the position, velocity, and acceleration of an object over time. Specifically, for an object in free fall, the acceleration is due to gravity and is constant. The most commonly used kinematic equation for finding the velocity of an object when its initial velocity and displacement are known is
\[ v^2 = u^2 + 2as \]
This equation tells us that the final velocity squared (\(v^2\)) equals the initial velocity squared (\(u^2\)) plus two times the acceleration (\(a\)) times the displacement (\(s\)). In our exercise, the object begins from rest, meaning that the initial velocity \(u\) is zero. Thus, the simplified form of this equation helps us calculate the impact velocity of an object in free fall. Understanding how to apply this equation correctly is crucial in making accurate calculations about the object's motion.
Free Fall Motion
The concept of free fall motion describes the movement of an object under the influence of gravitational force alone. It is an ideal condition where the only force acting on the object is gravity, and all other forces, such as air resistance, are neglected. In a free fall scenario, all objects, regardless of their mass, will have the same acceleration towards the Earth, known as the acceleration due to gravity. This constant acceleration is what allows us to use certain kinematic equations to solve for various parameters of motion, such as the impact velocity. When calculating impact velocity, it's essential to consider that in free fall, the initial velocity is zero when the object is simply dropped (as opposed to being thrown downwards), and the acceleration is always directed towards the Earth.
Gravity Acceleration
Gravity acceleration, commonly denoted as \(g\), is the acceleration that Earth imparts to objects on or near its surface. Its standard value is approximate \(9.81 m/s^2\), though it can slightly vary depending on geographical location and altitude. When objects are in free fall, gravity is the sole force acting upon them, causing a consistent acceleration downwards. This gravitational acceleration is a pivotal factor in calculating motion parameters such as displacement, velocity, and time during free fall. In our exercise, gravity acceleration is used to determine the impact velocity of the object. By plugging in the value of \(g\) into the kinematic equation, we get a direct relationship between the altitude from which the object was dropped and the velocity it will have upon impact, assuming no air resistance.

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