Calculating the Time for a Stone Thrown from a Roof to Hit the Ground

In this article, we will walk through a practical example of applying the principles of projectile motion to solve a real-world problem. This involves calculating the time it takes for a stone thrown from the rooftop of a 20-meter building to hit the ground when it is thrown at an initial speed of 30 m/s at a 45-degree angle. The process will involve breaking down the motion into horizontal and vertical components, and using the appropriate equations to determine the time.

Breakdown of the Problem

A stone is thrown from the rooftop of a 20-meter building at an initial velocity of 30 m/s at an angle of 45 degrees. We need to determine the time it takes for the stone to hit the ground.

Step 1: Determine the Initial Velocity Components

The initial velocity v_{0} is broken into horizontal v_{} and vertical v_{0y} components using trigonometric functions:

v_{} v_{0} cdot cos(theta) 30 cdot cos(45^circ) 30 cdot frac{sqrt{2}}{2} approx 21.21 text{m/s}

v_{0y} v_{0} cdot sin(theta) 30 cdot sin(45^circ) 30 cdot frac{sqrt{2}}{2} approx 21.21 text{m/s}

Here, theta is the angle of projection, which is 45 degrees.

Step 2: Set Up the Vertical Motion Equation

The vertical motion is influenced by gravity. The equation of motion for the vertical position y as a function of time t is:

y(t) y_{0} v_{0y} cdot t - frac{1}{2} g t^2

Where:

y_{0} 20 text{m} is the initial height g approx 9.81 m/s^2 is the acceleration due to gravity y(t) 0 is the final height when the stone hits the ground

Substituting the known values, we get:

0 20 21.21 t - frac{1}{2} 9.81 t^2

This simplifies to:

4.905 t^2 - 21.21 t - 20 0

Step 3: Solve for Time t

Using the quadratic formula to solve for t:

t frac{-b pm sqrt{b^2 - 4ac}}{2a}

Where a 4.905, b -21.21, and c -20:

t frac{21.21 pm sqrt{(-21.21)^2 - 4 cdot 4.905 cdot (-20)}}{2 cdot 4.905}

Calculating the discriminant:

sqrt{(-21.21)^2} 450.4641

4 cdot 4.905 cdot (-20) -392.4 (Subtracting a negative makes it positive):

450.4641 392.4 842.8641

Substituting back into the quadratic formula:

t frac{21.21 pm sqrt{842.8641}}{9.81}

Calculating the square root:

sqrt{842.8641} approx 29.06

Now, substituting:

t frac{21.21 pm 29.06}{9.81}

The two possible times are:

t frac{21.21 29.06}{9.81} approx frac{50.27}{9.81} approx 5.13 text{ seconds}

t frac{21.21 - 29.06}{9.81} approx frac{-7.85}{9.81} approx -0.80 text{ seconds}

The negative value is not physically meaningful.

Conclusion

The time it takes for the stone to hit the ground is approximately 5.13 seconds. This demonstrates the application of projectile motion principles and how the quadratic formula can be used to solve such problems.