A ball is dropped from a height x. It travels 122.5m in last second. Find x.

To find the height from which the ball was dropped, we can use the equations of motion under gravity. When an object is dropped, it accelerates downwards due to gravity, which we typically approximate as 9.81 m/s². The distance traveled by the ball in the last second of its fall can be calculated using a specific formula derived from the equations of motion.

Understanding the Problem

We know that the ball travels 122.5 meters during the last second of its fall. To find the total height \( x \), we can use the following steps:

Key Equations

1. The distance traveled in the last second can be expressed as:

• Distance in last second = \( s_n = u + \frac{1}{2} g (2n - 1) \)

Where:

• \( s_n \) = distance traveled in the nth second
• \( u \) = initial velocity (which is 0 for a dropped object)
• \( g \) = acceleration due to gravity (approximately 9.81 m/s²)
• \( n \) = total time in seconds

Setting Up the Equation

Since the ball is dropped, the initial velocity \( u = 0 \). Therefore, the formula simplifies to:

• \( s_n = \frac{1}{2} g (2n - 1) \)

We know that \( s_n = 122.5 \) m, so we can set up the equation:

• \( 122.5 = \frac{1}{2} \cdot 9.81 \cdot (2n - 1) \)

Solving for n

Now, let's solve for \( n \):

• \( 122.5 = 4.905 (2n - 1) \)
• \( 2n - 1 = \frac{122.5}{4.905} \)
• \( 2n - 1 \approx 24.94 \)
• \( 2n \approx 25.94 \)
• \( n \approx 12.97 \) seconds

Since time must be a whole number, we round \( n \) to 13 seconds. This means the ball takes approximately 13 seconds to hit the ground.

Calculating the Height x

Now that we have \( n \), we can find the total height \( x \) using the formula for the distance fallen:

• \( x = \frac{1}{2} g n^2 \)

Substituting \( n = 13 \) seconds:

• \( x = \frac{1}{2} \cdot 9.81 \cdot (13)^2 \)
• \( x = 4.905 \cdot 169 \)
• \( x \approx 828.45 \) meters

Final Result

Thus, the height from which the ball was dropped is approximately 828.45 meters. This calculation illustrates how we can apply the principles of kinematics to solve real-world problems involving free fall.