Question

A boulder falls off the top of a cliff during a storm. The cliff is 60 feet high. Find how long it will take for the boulder to hit the road below. Solve the falling object model for \(h=0\)

Short answer

It takes approximately 1.94 seconds for the boulder to hit the road.

Step-by-step solution

Understand the given problem

The problem involves the calculation of time when a boulder falls from a cliff of height 60 feet. It's a one-dimensional motion problem under constant acceleration due to gravity \(g = 32ft/s^2\).

Set up the equation

We can set up the equation based on the formula \(h = s - \frac{1}{2}gt^2\). The starting height \(s\) is 60 feet, the height \(h\) is 0 feet (as the boulder hits the ground), and \(g = 32ft/s^2\). Replace these values in the equation to get: \(0 = 60 - \frac{1}{2}(32)t^2\)

Solve the equation

Rearrange the equation to filter for \(t\). You get \(\frac{1}{2}(32)t^2 = 60\). Simplify it to \(16t^2 = 60\). Divide by 16 on both sides to get \(t^2 = \frac{60}{16}\). So, \(t = \sqrt{\frac{60}{16}} = \sqrt{3.75}\)

Calculate the square root

Calculate the square root \(\sqrt{3.75}\) using a calculator which approximates to 1.94 seconds. Therefore, this is the time it takes the boulder to fall. Note that since the time cannot be negative, we only consider the positive square root.

Key concepts

One-Dimensional Motion

One-dimensional motion refers to the movement of an object in a straight line. In such cases, the object's position changes over time along a single axis of motion, which is either vertical or horizontal.

For example, when a boulder falls off a cliff, it moves straight downward. This is a classic instance of one-dimensional motion, as the only direction that matters is the distance between the boulder and the ground below. When we analyze such motion, we typically ignore air resistance and other potential forces, focusing solely on the object's displacement, velocity, and acceleration along the one dimension.

Constant Acceleration

Constant acceleration occurs when an object's velocity changes at a consistent rate in a straight line. In the case of our falling boulder, the acceleration due to gravity is what propels it downwards. This acceleration is constant and does not change over time, making the motion predictable.

Mathematically, we express constant acceleration with the formula \(v = u + at\), where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is time. Since the starting velocity of the boulder is zero when it begins to fall, we simply use \(gt\) when substituting values into the kinematic equations for one-dimensional motion.

Gravity

Gravity is a natural phenomenon by which all things with mass are brought toward one another, including objects falling towards the Earth. On Earth, gravity gives weight to objects and causes them to fall towards the center of the planet when dropped.

The acceleration due to Earth's gravity is approximately \(9.8 m/s^2\) or \(32 ft/s^2\) and is denoted by the symbol \(g\). This value is significant, as it is used to calculate the fall time for any object near the Earth's surface, assuming no other forces act and taking into account the distance an object falls.

Quadratic Equations

Quadratic equations are mathematical expressions of the second degree, generally expressed as \(ax^2 + bx + c = 0\). These equations are notable in physics, especially when analyzing the motion of objects under constant acceleration. The solutions to a quadratic equation are found by factoring, completing the square, or using the quadratic formula.

In the context of a falling object, the equation that represents the object's height over time is a quadratic equation because the height depends on the square of the time. When we solve for time, we are aiming to find the point at which the height is zero, meaning the object has reached the ground. This involves simplifying and rewriting our original kinematic equation and then taking the square root to solve for time.