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. Which problem solving method do you prefer? Why?

Short answer

The time it takes for the boulder to hit the road is \(\sqrt{\frac{60}{0.5 \cdot 32}}\) seconds, or approximately 4.90 seconds.

Step-by-step solution

Identify the given variables and the unknown

The given values are: \[d = 60 feet \text{ and } g = 32 feet/sec^2\] The unknown to be found is the time \(t\).

Set up the equation

We will use the kinematic equation for uniformly accelerating bodies, \[d = 0.5 \cdot g \cdot t^2. \] Substituting the given values, we get \[60 = 0.5 \cdot 32 \cdot t^2.\]

Solve for the unknown variable

We rearrange the equation from step 2 to solve for the time \(t\): \[t^2 = \frac{60}{0.5 \cdot 32}\]. Taking the square root of both sides, we find \[t = \sqrt{\frac{60}{0.5 \cdot 32}}.\]

Key concepts

Understanding Free Fall

When discussing free fall, we refer to the motion of an object under the sole influence of gravity. This scenario assumes there is no air resistance affecting the object's descent. Free fall is a perfect example of uniformly accelerated motion because gravity imparts a constant acceleration on the object.

For instance, if a boulder falls from a cliff, the only force acting on it during its fall is the force of gravity. The acceleration due to gravity (\(g\)) is approximately 32 feet per second squared (or 9.8 meters per second squared if working with SI units). In free fall problems, like our example with the boulder, we calculate the time it takes for the object to reach the ground using kinematic equations which incorporate this constant acceleration.

Solving Quadratic Equations

Quadratic equations appear frequently in physics, particularly when discussing the trajectories of objects. These equations are polynomial equations of degree two, typically taking the form \( ax^2 + bx + c = 0 \), where \(a\text{, } b\text{, and } c\) are constants.

To solve a quadratic equation, we can use various methods such as factoring, completing the square, or applying the quadratic formula. For the boulder's motion, we arrived at a simplified quadratic equation where the time \( t \) is the variable to be determined. The equation was then solved by isolating \( t^2 \) and taking the square root of both sides to find the time it takes for the boulder to hit the ground.

Problem-Solving in Physics

Problem-solving in physics often involves identifying the relevant concepts, understanding the given information, formulating the right equations, and then executing mathematical manipulations to solve for the unknowns. In our cliff and boulder scenario, the process followed these steps:

Identifying the Variables

Here, the height from which the boulder fell and the acceleration due to gravity were given. The time was the unknown quantity to find.

Formulating Equations

We used the appropriate kinematic equation for objects under uniform acceleration to create a relationship between the known values and the unknown time.

Mathematical Solution

After substituting the known values into our equation, we performed algebraic manipulation to solve for time. It's essential to follow a logical sequence of steps and ensure that your mathematical operations are valid throughout the process.

Uniform Acceleration

Uniform acceleration occurs when an object's velocity changes at a constant rate. Gravity is a common source of uniform acceleration which is characteristic of free-falling objects.

In the context of the boulder problem, the force of gravity caused the boulder to accelerate towards the Earth at a constant rate (\(g = 32 feet/sec^2\)). In kinematic equations, this constant acceleration simplifies our calculations, as we can predict the position and velocity of the object at any given time.

Knowing that the acceleration is constant allows us to use the basic kinematic equation \( d = 0.5 \times g \times t^2 \) to correlate the distance fallen, the acceleration due to gravity, and the time the object has been falling. This is why our calculation for the boulder only needed these specific variables to determine the time to impact.