Question

Bill Jones has a bad night in his bowling league. Feeling disgusted when he gets home, he drops his bowling ball out the window of his apartment, from a height of \(63.17 \mathrm{~m}\) above the ground. John Smith sees the bowling ball pass by his window when it is \(40.95 \mathrm{~m}\) above the ground. How much time passes from the time when John Smith sees the bowling ball pass his window to when it hits the ground?

Short answer

Answer: \(Δt = \sqrt{\frac{2(63.17-40.95)}{9.81}} - \sqrt{\frac{2(40.95)}{9.81}}\)

Step-by-step solution

Determine the height difference of the fall

Calculate the height difference from where John saw the ball to the ground. The height difference is: \(Δh = 63.17 \mathrm{~m} -40.95 \mathrm{~m}\)

Calculate the time from John's window to the ground

Now we need to find the time it takes for the bowling ball to fall from 40.95 m to the ground. Use the free fall equation: \(Δh = \frac{1}{2}gt^2\) Where \(g\) is the acceleration due to gravity (\(9.81 \mathrm{~m/s^2}\)) and \(t\) is the time. Solve for \(t\): \(t=\sqrt{\frac{2Δh}{g}}\) Plug the values in and calculate the time: \(t=\sqrt{\frac{2(63.17-40.95)}{9.81}}\)

Calculate the time it takes for the bowling ball to fall into John's view

Now we need to find the time it takes for the bowling ball to fall from 63.17 m to John's view (40.95 m). Since it's also a free fall, we can use the same equation: \(t_2=\sqrt{\frac{2(40.95)}{9.81}}\)

Calculate the time difference

To find the time it takes for the bowling ball to pass John's window to when it hits the ground, calculate the difference between the two times found in Step 2 and Step 3: \(Δt = t - t_2\) Finally, plug the values in and calculate the time difference: \(Δt = \sqrt{\frac{2(63.17-40.95)}{9.81}} - \sqrt{\frac{2(40.95)}{9.81}}\) The result will give us the time that passes from when John Smith sees the bowling ball pass his window to when it hits the ground.

Key concepts

Acceleration Due to Gravity

Acceleration due to gravity is a constant force that pulls all objects towards the center of the Earth when they are in free fall. This acceleration is denoted as \( g \) and has a standard value of \( 9.81 \text{ m/s}^2 \) on the surface of the Earth. It's crucial to understand that this acceleration is the same for all objects, regardless of their mass, which means both a feather and a bowling ball would accelerate at the same rate if there were no air resistance. In our bowling ball problem, the knowledge of \( g \) allows us to predict how long the ball will take to fall from a certain height.

Kinematic Equations

Kinematic equations describe the motion of objects in one dimension when the acceleration is constant, which is exactly the case during free fall. The particular equation we use for free-fall problems is \( Δh = \frac{1}{2}gt^2 \), relating the height \( Δh \), acceleration due to gravity \( g \), and time \( t \) of fall. These equations are vital for solving problems like the one involving our bowling ball, as they give us a mathematical model to work out how different variables affect the motion of the ball as it falls.

Time of Fall Calculation

To calculate the time of fall in a free fall scenario, we often rearrange the kinematic equation to solve for time \( t \). In this particular case, we use the formula \( t = \sqrt{\frac{2Δh}{g}} \) to find the time it takes for the bowling ball to fall a certain distance. By plugging in the difference in height and the acceleration due to gravity, we are able to compute the time interval during which the ball travels past John's window to the ground. The calculated time is crucial for understanding how objects accelerate under the influence of gravity over time.

Motion in One Dimension

Motion in one dimension refers to the movement of an object along a straight line. This linear motion can be described using kinematic equations, assuming the acceleration is uniform. In our example's context, the bowling ball exhibits motion in one dimension as it moves straight down from Bill's window to the ground. Understanding motion in one dimension is essential for predicting how an object will move when only one force acts upon it, such as gravity in the case of free fall.