Question

Refer to the falling object of Example 1 on page 457 , where we found that the total distance, in feet, that an object falls in \(n\) seconds is given by \(S_{n}=16 n^{2}\). (a) Find the total distance that an object falls in 7,8,9 seconds. (b) If the object falls from 1000 feet at time \(t=0\) Calculate its height at \(t=7, t=8, t=9 \mathrm{sec}-\) onds. (c) Does part (b) make sense physically?

Short answer

#Problem# An object falls from a height of 1000 feet, and its distance fallen after n seconds is given by the formula: \(S_n = 16n^2\) (a) Find the total distance fallen after 7, 8, and 9 seconds. (b) Calculate the height of the object above the ground at these times. (c) Analyze whether your answers to part (b) make physical sense. #Answer# (a) Total distance fallen after 7, 8, and 9 seconds: 784 feet, 1024 feet, and 1296 feet, respectively. (b) Height of the object above the ground at 7, 8, and 9 seconds: 216 feet, -24 feet, and -296 feet, respectively. (c) The height after 7 seconds makes physical sense since the object is still above the ground. However, the calculated heights after 8 and 9 seconds are negative, which does not make physical sense, as the object should have stopped at ground level (0 feet) after falling from a height of 1000 feet. The discrepancy might be due to the constant acceleration model not considering the object's stopping once it reaches the ground.

Step-by-step solution

(a) Find the total distance fallen)

To find the total distance an object falls in 7, 8, and 9 seconds, use the given formula \(S_n = 16n^2\) and substitute \(n\) with the respective time values.

(7 seconds)

Substitute \(n = 7\) in the formula: \(S_7 = 16(7^2) = 16(49) = 784\) feet

(8 seconds)

Substitute \(n = 8\) in the formula: \(S_8 = 16(8^2) = 16(64) = 1024\) feet

(9 seconds)

Substitute \(n = 9\) in the formula: \(S_9 = 16(9^2) = 16(81) = 1296\) feet

(b) Calculate the height)

To find the height of the object at \(t = 7, 8\) and \(9\) seconds, subtract the total distance fallen (\(S_n\)) at each of these times from the initial height of 1000 feet.

(7 seconds)

Height at \(t = 7\) seconds: \(1000 - S_7 = 1000 - 784 = 216\) feet

(8 seconds)

Height at \(t = 8\) seconds: \(1000 - S_8 = 1000 - 1024 = -24\) feet

(9 seconds)

Height at \(t = 9\) seconds: \(1000 - S_9 = 1000 - 1296 = -296\) feet

(c) Analyzing the physical sense)

The height after 7 seconds (\(216\) feet) makes physical sense since the object is still above the ground. However, at 8 seconds (\(-24\) feet) and 9 seconds (\(-296\) feet), the calculated height is negative, which means that the object is below ground level. This does not make physical sense, as the object should have stopped at ground level (0 feet) after falling from a height of 1000 feet. This discrepancy may be due to the constant acceleration model not taking into account the impact and stopping of the object once it reaches the ground.

Key concepts

Distance Formula

In physics, the distance formula provides a powerful tool to calculate how far an object has moved in a particular scenario. In the case of a falling object, the distance it travels can be expressed using a formula that reflects the acceleration due to gravity. Simply put, the distance formula for a falling object beginning from rest in Earth's gravity can be stated as \( S_n = 16n^2 \), where \( S_n \) is the total distance fallen in feet, and \( n \) is the time in seconds.
Utilizing this formula, you can determine how far an object will fall over a given period, specifically 7, 8, or 9 seconds in the exercise you are dealing with. This formula assumes the absence of air resistance and other forces, focusing solely on the object and gravity's influence.

• At 7 seconds, substituting into the formula yields a distance of 784 feet.
• At 8 seconds, the object falls 1024 feet in total.
• Finally, at 9 seconds, it falls 1296 feet.

Gravity

Gravity is the natural phenomenon by which objects with mass attract one another. On Earth, it causes objects to fall toward the ground when dropped. This constant force affects everything on the surface of the planet, pulling it toward the center. Earth's gravitational acceleration is approximately 32 feet per second squared. However, because the formula used in the exercise simplifies calculations, we use half of that value, which is 16, to represent half the squared gravitational acceleration.
In the context of the falling object exercise, gravitational acceleration is the sole cause of the object's increased velocity as it falls. With each passing second, gravity helps the object gain speed, accelerating uniformly and resulting in increased displacement over time. Air resistance and terminal velocity are ignored in these calculations to maintain simplicity. Gravity is fundamental to kinematic equations, making it pivotal when determining an object’s motion under its influence.

Kinematics

Kinematics is a branch of physics concerned with the motion of objects without considering the forces causing the motion. The equations of kinematics allow us to evaluate an object's position, velocity, and acceleration at various points in time.In this exercise, kinematics is applied using a quadratic function: \( S_n = 16n^2 \), where changes in height result from accelerating due to gravity over a period.
This quadratic relationship indicates a direct proportionality between the square of time (\( n^2 \)) and the distance (\( S_n \)) that the object falls. This means as time progresses, the distance fallen increases exponentially.Due to the simplified nature of kinematic equations, they assume ideal conditions, such as no air resistance and a starting velocity of zero.Actual scenarios might require more complex calculations to account for external factors.

Initial Height

Initial height refers to the starting point of the object above a reference level, commonly the ground. In this exercise, the object begins its descent from 1000 feet. The initial height is critical because it sets the point from which we calculate how far an object will fall or remain above the ground.
To determine the height of the object at different times (7, 8, and 9 seconds), you subtract the distance fallen from the initial height:

• At 7 seconds, the height is 216 feet, as 784 feet has been covered.
• At 8 seconds, the height calculation shows the object at -24 feet, indicating it passed the ground level.
• At 9 seconds, it registers at -296 feet, reinforcing that the negative value denotes a physical impossibility in real-life scenarios.

Negative results symbolize that physical constraints like the ground level have been bypassed. The exercise illustrates the importance of understanding numerical solutions' real-life applicability.