Chapter 1: Problem 15

How does the velocity of a freely falling body change with time? How does the distance it has fallen change? How about the acceleration?

Short Answer

Expert verified

Answer: The velocity of a freely falling body increases linearly with time, with a slope representing the acceleration due to gravity (-g). The distance covered by the falling object varies as the square of the time elapsed, with a factor of -0.5g representing the gravitational acceleration. The acceleration of the object remains constant at -g throughout its fall, indicating that it is influenced only by gravitational force.

Step by step solution

Identify the equations of motion

For a body moving with constant acceleration, we have three main equations of motion. Let v be the final velocity, u the initial velocity, a the constant acceleration, s the distance traveled, and t the time passed. 1. \(v = u + at\) 2. \(s = ut + \frac{1}{2}at^2\) 3. \(v^2 = u^2 + 2as\) Since the body is freely falling, we'll use the acceleration due to gravity, which is denoted as 'g'. The acceleration will be negative if we consider the upward direction to be positive (falling downwards is the negative direction).

Determine expressions for velocity, distance, and acceleration

The object starts falling from rest, so its initial velocity u = 0. Let's substitute this value and the acceleration due to gravity (a = -g) into the equations of motion. 1. Velocity: \(v = 0 - gt = -gt\) 2. Distance: \(s = 0t - \frac{1}{2}gt^2 = - \frac{1}{2}gt^2\) 3. For the acceleration, we already know that it's a constant due to gravity, hence the acceleration is a = -g.

Interpret the results

The expressions we obtained show the following relationships between the falling body and time: 1. Velocity: As time goes on, the velocity of the falling object increases linearly with time, with a slope representing the acceleration due to gravity (-g). The object's velocity will be negative in the downward direction. 2. Distance: The distance covered by the falling object varies as the square of the time elapsed, with a factor of -0.5g representing the gravitational acceleration. This means that the object falls a greater distance every second. 3. Acceleration: The acceleration of the object remains constant at -g throughout its fall, indicating that it is influenced only by gravitational force.

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How does the velocity of a freely falling body change with time? How does the distance it has fallen change? How about the acceleration?

Short Answer

Step by step solution

Identify the equations of motion

Determine expressions for velocity, distance, and acceleration

Interpret the results

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