Problem 11 - Applied Physics

Question

A ball is thrown downward from the top of a building at a speed of \(75 \mathrm{ft} / \mathrm{s}\). Find its velocity as it hits the ground \(475 \mathrm{ft}\) below. Disregard air resistance.

Step-by-Step Solution

Answer

The velocity is approximately 190.32 ft/s.

Step 1: Identify Initial Conditions

The initial velocity \( v_0 \) of the ball when thrown is given as \( 75 \, \text{ft/s} \). The initial position \( h_0 \) is at the top of the building, which we'll consider as the zero point, and the ball falls a total distance of \( 475 \, \text{ft} \) down.

Step 2: Understand the Equation of Motion

Since the ball is in freefall apart from the initial velocity, we need to consider the acceleration due to gravity, which is approximately \( 32.2 \, \text{ft/s}^2 \). The vertical motion of the ball follows the equation: \[ v^2 = v_0^2 + 2g d \] where \( v \) is the final velocity, \( v_0 \) is the initial velocity, \( g \) is the acceleration due to gravity, and \( d \) is the distance fallen.

Step 3: Plug in Known Values

Substitute the values into the equation: \[ v^2 = (75)^2 + 2(32.2)(475) \] Calculate the terms separately:- \( v_0^2 = 75^2 = 5625 \)- \( 2gd = 2(32.2)(475) = 30595 \)Adding these gives:\[ v^2 = 5625 + 30595 = 36220 \]

Step 4: Solve for Final Velocity

Take the square root of both sides of the equation to find \( v \):\[ v = \sqrt{36220} \approx 190.32 \, \text{ft/s} \] Therefore, the velocity of the ball as it hits the ground is approximately \( 190.32 \, \text{ft/s} \).

Key Concepts

Initial Conditions

When solving kinematics problems, especially those involving objects in motion under the influence of gravity, identifying the initial conditions is crucial. In our exercise, the ball is thrown downward. This gives us an initial velocity. In this case, the initial velocity \( v_0 \) is provided as \( 75 \, \text{ft/s} \). Always remember:

• The initial velocity is the speed at which the object begins its journey.
• The initial position \( h_0 \) is typically your reference point. Here, it’s the top of the building, marked as zero.

Knowing these conditions lays the groundwork for further calculations. The initial conditions define how to apply the laws of motion and are vital in predicting future positions or velocities of objects in kinematics problems. When you have these values, you can use equations to find out how the motion will develop.

Equation of Motion

The equation of motion is a fundamental tool in kinematics. It relates to the initial velocity \( v_0 \), the final velocity \( v \), the acceleration \( g \), and the distance \( d \). The specific equation here is:

\[ v^2 = v_0^2 + 2gd\]This equation helps calculate the final velocity of the object once it has traveled a certain distance under constant acceleration. Here’s what each term means:

• \( v_0^2 \) is the initial velocity squared.
• \( 2gd \) is twice the product of gravitational acceleration \( g \) and the distance fallen \( d \).
• \( v^2 \) is the final velocity squared.

By breaking it down:- Identify and plug in the known values.- Solve for the unknown, usually the final velocity \( v \).This equation incorporates both the effects of initial speed and how much the object is accelerated over the distance it covers.

Acceleration Due to Gravity

Gravity is a natural force that applies to objects in free fall. When you hear about objects falling, it's crucial to understand the role of acceleration due to gravity. Typically, this value is \( 32.2 \, \text{ft/s}^2 \) on Earth's surface.

Gravity accelerates objects downward. This means every second, the object's speed increases by \( 32.2 \, \text{ft/s} \). Here are some key points about gravitational acceleration:

• It's a constant force that acts in the downward direction.
• It only affects the vertical component of movement, not horizontal movement.
• Regardless of initial speed, gravity will increase the velocity of a falling object continuously as long as it's in free fall.

In the context of our exercise, the consistent acceleration due to gravity impacts how fast the ball travels the \( 475 \, \text{ft} \) before hitting the ground, influencing the final velocity of the ball—calculated using the equation of motion.