Problem 54 - Calculus with Concepts in Calculus

Question

In order to discover the height above ground of a ten-story apartment, Pat drops a ball from its balcony. If the ball hits the grass below after \(2.5\) seconds, determine the height (in meters) from which the ball was dropped.

Step-by-Step Solution

Answer

The height is 30.625 meters.

Step 1: Identify the Known Values

First, we need to identify the values that have been given. The time, \(t\), is provided as \(2.5\) seconds. We will assume that the only force acting on the ball is gravity, and the acceleration due to gravity, \(g\), is \(9.8\text{ m/s}^2\).

Step 2: Use the Free Fall Distance Formula

The height, \(h\), from which the ball is dropped can be calculated using the free fall distance formula: \[ h = \frac{1}{2}gt^2 \]This formula comes from the kinematic equations, assuming initial velocity \(v_0 = 0\).

Step 3: Substitute the Known Values into the Formula

Substitute \(g = 9.8\text{ m/s}^2\) and \(t = 2.5\) seconds into the formula:\[ h = \frac{1}{2} \times 9.8 \times (2.5)^2 \]

Step 4: Perform the Calculations

Calculate the square of the time:\( (2.5)^2 = 6.25 \)Then multiply by 9.8 and then by 0.5:\[ h = 0.5 \times 9.8 \times 6.25 = 30.625 \]

Step 5: Interpret the Result

The calculated height, \(h\), is \(30.625\) meters.

Key Concepts

Kinematic Equations

Kinematic equations are mathematical formulas that describe the motion of objects under constant acceleration. In physics, they help us to predict and understand the behavior of moving objects. These equations link different physical quantities such as displacement, velocity, acceleration, and time. For an object in free fall, we primarily use two kinematic equations:

• Displacement equation: \( s = v_0t + \frac{1}{2}at^2 \)
• Velocity equation: \( v = v_0 + at \)

The displacement equation is mainly used for calculating the distance an object has fallen in free fall, assuming zero initial velocity, which simplifies it to:\[ s = \frac{1}{2}gt^2 \]where:- \( s \) is the displacement (distance fallen)- \( g \) is the acceleration due to gravity- \( t \) is the time elapsedThese equations are powerful tools in mechanics, simplifying the process of analyzing motion, especially when dealing with the effects of gravity.

Gravity Calculation

Gravity plays a crucial role in free fall physics, dictating how fast objects accelerate toward Earth when dropped. The acceleration due to gravity is denoted by \( g \), with a standard value of \( 9.8 \text{ m/s}^2 \) on Earth's surface. This value is consistent at small distances from the Earth's surface but can change slightly depending on location and altitude.
In calculations involving free fall, it's essential to remember:

• Gravity is always directed downward.
• Gravity causes a constant acceleration regardless of the mass of the object being dropped.
• It simplifies many calculations because it makes the initial velocity (\( v_0 \)) equal to zero when an object is simply dropped.

The concept of gravitational acceleration simplifies the prediction of free-falling objects, allowing us to use kinematic equations to find how far or how fast an object falls within a specific time frame.

Motion Under Gravity

Motion under gravity refers to the path and behavior an object follows when influenced only by gravity, with no other forces acting on it. Analyzing this motion helps us understand everyday phenomena like falling leaves or a ball dropping from a height.When objects are in free fall, they experience motion only due to gravity, leading to constant acceleration. This means the velocity of the object constantly increases until it hits the ground. Here, the object starts with an initial velocity of zero if released from rest.
Key points when considering motion under gravity:

• Initial velocity often starts from zero when dropped directly.
• Acceleration remains constant at \( 9.8 \text{ m/s}^2 \) downward.
• The only force considered is the gravitational force.

Understanding motion under gravity enables us to predict outcomes in real-life situations. It allows physicists and engineers to calculate trajectories and impact scenarios essential in designing safe and functional environments.