Question

Suppose a rock falls from rest from a height of 100 meters and the only force acting on it is gravity. Find an equation for the velocity \(v(t)\) as a function of time, measured in meters per second.

Short answer

The equation for the velocity is \(v(t) = 9.81t\).

Step-by-step solution

Understand the Problem

The problem involves a rock falling from a height of 100 meters. Since it falls freely under gravity, we need to find the velocity as a function of time. The initial velocity is zero because it falls from rest.

Identify the Relevant Equations

The object falls under the influence of gravity alone, which means we can apply the kinematic equations. For an object starting from rest and accelerating due to gravity, the velocity at any time t is given by the equation:\[ v(t) = v_0 + g \cdot t \]where \(v_0\) is the initial velocity and \(g\) is the acceleration due to gravity.

Plug in Values

The initial velocity \(v_0\) is zero (since the rock starts from rest), and the acceleration due to gravity \(g\) is approximately 9.81 m/s². Plug these values into the equation:\[ v(t) = 0 + 9.81 \cdot t = 9.81t \]

Write the Final Equation

The equation for the velocity of the rock as a function of time is obtained. It indicates that velocity increases linearly over time under constant acceleration.\[ v(t) = 9.81t \]

Key concepts

Velocity

Velocity tells us how fast an object is moving and in which direction. It is a vector quantity, meaning it has both magnitude and direction. When solving problems in kinematics, identifying how velocity changes over time is crucial.

• Magnitude: The speed of the object, telling us how fast it is going regardless of direction.
• Direction: The path in which the object moves, described using a specific direction.

For an object falling freely from rest, like our rock from the exercise, the initial velocity is zero because there is no initial motion. As time passes, velocity increases due to gravitational acceleration. The relation between velocity and time is linear if the acceleration is constant, leading to equations such as:\[ v(t) = v_0 + g \cdot t \]where \(v_0\) is the starting velocity. However, in free fall from rest, \(v_0 = 0\), simplifying to:\[ v(t) = g \cdot t \]

Acceleration Due to Gravity

Acceleration due to gravity is a constant force that pulls objects towards the center of the Earth. It is usually denoted by \(g\) and has an average value of approximately 9.81 m/s² on Earth. This acceleration is pivotal in kinematics as it affects how velocity and position change over time.

• Nature of \(g\): It is a constant value that acts in the same way on all objects, regardless of their mass.
• Impact on Velocity: The longer an object is under this acceleration, the faster and further it will travel downward.

In the exercise, the rock’s acceleration remains constant due to gravity alone. We used this constant \(g\) to predict the rock's velocity over time,leading to the direct relationship described by:\[ v(t) = 9.81t \]This indicates that each second, the velocity increases by 9.81 meters per second.

Free Fall

Free fall describes an object's motion when gravity is the only force acting upon it. In this scenario, no other forces, like air resistance, play a role. The concept is essential in understanding motions under gravity without external interference.

• Characteristics: The only force involved is gravity, causing the object to accelerate at a rate of \(g\).
• Ideal Condition: In reality, air resistance might slightly affect the motion, but in most basic physics problems, it is neglected for simplicity.

In the given exercise, the rock undergoes free fall from a 100-meter height. We calculated its velocity as a linear function of time:\[ v(t) = 9.81t \]This linearity holds because free fall assumes only gravity influences the rock, allowing us to focus on gravitational effects without other complicating factors like wind or drag. The simplicity of free fall helps in learning foundational kinematic principles.