Question

A steel ball is dropped from a height of \(12.37 \mathrm{~m}\) above the ground. What is its speed when it reaches \(2.345 \mathrm{~m}\) above the ground?

Short answer

Answer: The speed of the steel ball when it reaches 2.345 meters above the ground is approximately \(14.01 \mathrm{~m/s}\) downward.

Step-by-step solution

Calculate the displacement

Since the ball drops from a height of \(12.37 \mathrm{~m}\) to a height of \(2.345 \mathrm{~m}\), the displacement (\(s\)) is \(12.37 - 2.345 = 10.025 \mathrm{~m}\).

Identify the initial velocity

The ball is dropped from rest, so the initial velocity (\(u\)) is \(0 \mathrm{~m/s}\).

Identify the acceleration due to gravity

The acceleration due to gravity (\(a\)) is approximately \(9.81 \mathrm{~m/s^2}\) downward. Since the ball is falling downward, we can keep \(a\) as positive.

Use the equation of motion to find the final velocity

We will use the equation \(v^2 = u^2 + 2as\) to find the final velocity (\(v\)) of the ball. Plugging in our values, we get: \(v^2 = 0^2 + 2(9.81)(10.025)\) \(v^2 = 196.4135\) Now, we take the square root to find \(v\): \(v = \sqrt{196.4135}\) \(v \approx 14.01 \mathrm{~m/s}\) So, the speed of the steel ball when it reaches \(2.345 \mathrm{~m}\) above the ground is approximately \(14.01 \mathrm{~m/s}\) downward.

Key concepts

Motion Under Gravity

When we talk about motion under gravity, we are referring to how objects move when the only force acting on them is gravity. This is commonly observed when an object is dropped and allowed to fall freely towards the Earth. In this situation, gravity acts as a constant acceleration directed towards the center of the Earth.

The force of gravity causes all freely falling objects to accelerate downward at the same rate, regardless of their mass. This acceleration is denoted by the symbol 'g' and has a value of approximately \(9.81 \text{m/s}^2\) on the surface of the Earth.

How Gravity Affects Motion

Gravity imparts a steady increase in velocity in the direction of the force. If an object starts from rest, its velocity will increase linearly with time as long as it is in free fall. The relationship between time, velocity, and acceleration due to gravity can be determined using kinematic equations, which provide useful tools to solve problems involving motion under gravity.

Equations of Motion

The equations of motion are a set of formulas that describe the relationship between displacement, velocity, acceleration, and time for objects moving with uniform acceleration, including free-falling objects. Among these equations is the formula \(v^2 = u^2 + 2as\), which is essential for solving the provided exercise.

Let's break down the equation: \(v\) represents the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(s\) is the displacement of the object. These equations of motion are rooted in basic principles of Newtonian mechanics and are incredibly powerful for analyzing the motion of objects under a variety of conditions.

Applying the Equation

Using the proper equation of motion allows us to find an object's final velocity without the need to measure the time involved. This is particularly helpful when we have constant acceleration, like that of gravity, and known displacement.

Free Fall

Free fall is the motion of an object under the influence of gravitational force only. It's an idealized situation in which air resistance is neglected, meaning the object is not subjected to any form of air drag. In this scenario, all objects fall at the same acceleration \(g\), irrespective of their masses.

The experience of free fall is uniform for all objects because it is solely dependent on the force of gravity. When we say an object is in free fall, it is understood that the initial velocity is zero if it is simply dropped, not thrown, from a height.

Characteristics of Free Fall

In problems like the one we're discussing, we'll assume that the only force acting on the falling object is gravity, simplifying our calculations as we can ignore other forces such as air resistance. This assumption allows us to predict the velocity and position of a freely falling object at any point during its descent.

Initial Velocity

Initial velocity is the velocity of an object before it begins to experience acceleration. In our example, the steel ball's initial velocity \(u\) is zero because it is dropped from rest, not thrown with any initial speed. The concept of initial velocity is crucial in kinematics, as it serves as the starting point for calculating changes in velocity due to acceleration.

In free fall, the only acceleration acting on the object is due to gravity, so understanding its initial state is key to determining how its velocity will change over time. Additionally, initial velocity is a vector quantity—it has both magnitude and direction—meaning that in free fall, the direction of initial velocity is vertically downward, the same as the direction of acceleration.

Significance in Equations of Motion

The initial velocity can significantly influence an object's trajectory and final speed. By accurately accounting for it, we can apply the kinematic equations to predict subsequent motion accurately, such as in the textbook exercise where initial velocity is a fundamental component of the final solution.