Problem 99 - Fundamentals of Physics

Question

A ball is thrown vertically downward from the top of a \(36.6-\mathrm{m}\) -tall building. The ball passes the top of a window that is \(12.2 \mathrm{~m}\) above the ground \(2.00 \mathrm{~s}\) after heing thrown. What is the specd of the ball as it passes the top of the window?

Step-by-Step Solution

Answer

The speed of the ball as it passes the top of the window is 22.01 m/s.

Step 1: Identify Known Values

We have the initial height of the building as 36.6 m, the window's height above the ground as 12.2 m, and the time taken to reach this window height is 2.00 s. The acceleration due to gravity, g, is approximately 9.81 m/s² (downward).

Step 2: Calculate Initial Drop Distance

Calculate the distance the ball drops to reach the top of the window: 36.6 m (building height) - 12.2 m (window height) = 24.4 m. This is the distance the ball has traveled in the vertical direction.

Step 3: Use Kinematic Equation for Final Velocity

To find the final speed as the ball passes the window, use the kinematic equation:\[ v = u + gt \]where:- \( u \) is the initial velocity (which we need to find first, if not given explicitly as 0 m/s due to 'thrown' terminology),- \( g = 9.81 \text{ m/s}^2 \) (acceleration due to gravity),- \( t = 2.00 \text{ s} \) (time).

Step 4: Solve for Final Velocity

Given that the ball was thrown downward, let's assume the ball started with an initial speed \( u \), and solve the equation:\[ 24.4 = ut + \frac{1}{2}gt^2 \]\[ 24.4 = 2u + \frac{1}{2}(9.81)(2^2) \]\[ 24.4 = 2u + 19.62 \]\[ 2u = 24.4 - 19.62 \]\[ 2u = 4.78 \]\[ u = 2.39 \text{ m/s} \]Now, plug \( u = 2.39 \text{ m/s} \) into the first kinematic equation:\[ v = 2.39 + 9.81 imes 2 \]\[ v = 2.39 + 19.62 \]\[ v = 22.01 \text{ m/s} \]

Key Concepts

Projectile motion

Projectile motion occurs when an object is launched into the air and moves under the influence of gravity. In the exercise, the ball is thrown vertically downward, a specific case of projectile motion where the motion is confined to a single vertical direction. Concepts of projectile motion include:

• Initial Velocity: The speed and direction at which an object is initially thrown. In this context, initial velocity is downward and calculated as part of the problem.


• Vertical Motion: As gravity is the only force acting, the main focus is on understanding how the ball travels downwards.


• Kinematic Equations: These equations relate various quantities like velocity, acceleration, time, and displacement to analyze motion. In our exercise, they help determine the speed of the ball as it passes the window.

Free fall

Free fall describes any motion of a body where gravity is the only force acting upon it. Even though the ball in the exercise is thrown initially, once released, it effectively enters a free fall condition.
Understanding this simplifies calculations as the only acceleration acting is due to gravity.

• Constant Acceleration: In free fall, the acceleration of the object is constant because it’s only affected by gravity, which is approximated as 9.81 m/s².


• Vertical Displacement: This is calculated to determine how far the ball falls in a given time. It's a key step in calculating its speed at a particular point.

During free fall, the object's speed increases due to gravitational acceleration, a factor important in determining the velocity of the ball as it passes the window.

Acceleration due to gravity

Acceleration due to gravity is essential in analyzing any motion involving free fall or projectile motion. This acceleration impacts how quickly an object speeds up as it falls.
Gravity is always acting downward, and it influences:

• Speed: As seen in the exercise, the ball's speed increases as it falls. Calculation of speed incorporates gravity's influence over time.


• Kinematic Equations: These rely heavily on gravity to calculate various motion components, like the final velocity or distance traveled.


• Uniform Force: Gravity is a constant force, meaning its effect is predictable and can be precisely included in calculations.

In our exercise, gravity's acceleration of 9.81 m/s² helps us determine both the initial velocity and speed as the ball passes the window, highlighting its integral role in this and many other kinematics problems.