Question

At a construction site a pipe wrench strikes the ground with a speed of \(24.0 \mathrm{~m} / \mathrm{s}\). (a) From what height was it inadvertently dropped? (b) For how long was it falling?

Short answer

So, the wrench was dropped from a height of approximately 29.4 meters and it was falling for approximately 2.45 seconds.

Step-by-step solution

Analyzing Part (a)

We start from part (a). We want to know from what height the wrench was dropped. From the problem, we know the final speed of the wrench is 24.0 m/s when reaching the ground. We can use the formula from physics \(v^2 = u^2 + 2gh\), where \(v\) is the final speed, \(u\) is the initial speed, \(g\) is acceleration due to gravity, and \(h\) is the height. Since the wrench is dropped, the initial speed \(u\) is 0. Plugging these values in, we get \(h = \frac{v^2}{2g}\). We know \(v = 24.0 m/s\) and \(g = 9.8 m/s^2\).

Calculating the height

Plugging \(v\) and \(g\) into the equation gives \(h = \frac{(24.0 m/s)^2}{2*9.8 m/s^2} = 29.4 m\). Thus, the wrench was dropped from a height of 29.4 meters.

Analyzing Part (b)

Now, to calculate how long was it falling, we can use the second equation of motion \(v = u + gt\), where \(v\) is the final velocity, \(u\) is the initial velocity, \(g\) is acceleration due to gravity, and \(t\) is the time. From the problem, we know the final velocity \(v\) is 24.0 m/s and since the wrench is dropped, the initial velocity \(u\) is 0. The acceleration due to gravity \(g\) is 9.8 m/s^2. Now rearrange this equation to solve for \(t\): \(t = \frac{v-u}{g}\).

Calculating the time

Plugging \(u\), \(v\), and \(g\) into the equation gives \(t = \frac{24.0 m/s - 0}{9.8 m/s^2} = 2.45 s\). Thus, the wrench was falling for approximately 2.45 seconds.

Key concepts

Free Fall Physics

When objects fall due to gravity with no other forces acting on them, such as air resistance, they are said to be in free fall. Free fall is a motion where the only force acting upon an object is gravity. The simplicity of free fall allows us to model the motion mathematically with precision.

An important characteristic of free fall is that all objects, regardless of their mass, fall at the same rate when air resistance is negligible. This concept was famously demonstrated by Galileo at the Leaning Tower of Pisa. In the case of the pipe wrench from our exercise, we assume that it is in a state of free fall, meaning it has constant acceleration downward due to gravity and no initial velocity when it was dropped.

Equations of Motion

The equations of motion describe the relationship between an object's displacement, velocity, acceleration, and time. In the scenario of the falling pipe wrench, we deal with two of these equations.

The first is \(v^2 = u^2 + 2gh\), which relates the final velocity \(v\) of an object to its initial velocity \(u\), the height \(h\) from which it falls, and the acceleration due to gravity \(g\). This equation is derived from the work-energy principle and can determine the height from which the wrench was dropped by using its final velocity.

The second equation, \(v = u + gt\), is used for calculating the time an object is in free fall. It connects final velocity \(v\), initial velocity \(u\), acceleration due to gravity \(g\), and time \(t\). This kinematic equation is rooted in the concept of constant acceleration and allows us to find out how long the wrench was falling before it hit the ground.

Acceleration due to Gravity

The acceleration due to gravity, usually denoted as \(g\), is the acceleration that Earth imparts to objects on or near its surface. The standard value of \(g\) on Earth's surface is \(9.8 \mathrm{m/s}^2\), though this can vary slightly depending on altitude and latitude.

In our calculations involving the falling pipe wrench, we use \(g\) as a constant to find out the height and the time the wrench was in free fall. It's essential to understand that \(g\) not only accelerates objects down towards the Earth but also dictates the trajectory and speed of an object's fall, as it is the only force acting on an object in free fall conditions. The universal nature of \(g\) simplifies the problem greatly, as it applies uniformly to any object, large or small.