Example 2 A student throws a book horizontally out a dorm window with a speed of 12.5 m/s. The book lands on the ground 31.8 m from the base of the building. How high is the window above the ground?
To solve the problem, we can use the equations of motion for a projectile. We know that the book is thrown horizontally, so the initial vertical velocity is zero. We can use the following equations:
d = V₀t + 1/2at²
Vf = V₀ + at
where d is the vertical distance traveled, V₀ is the initial vertical velocity, a is the acceleration due to gravity (-9.8 m/s²), t is the time of flight, and Vf is the final vertical velocity (which is also zero at the highest point of the trajectory).
We can use the second equation to find the time of flight:
Vf = V₀ + at
0 = 0 - 9.8t
t = 0 s
This tells us that the book spends no time in the air at the highest point of its trajectory, since it has no initial vertical velocity. Therefore, the vertical distance traveled is simply the height of the window above the ground:
d = 1/2at²
d = 1/2(-9.8 m/s²)(t²)
d = 1/2(-9.8 m/s²)(0 s)²
d = 0 m
So the window is at a height of 0 meters above the ground. This may seem counterintuitive, since the book clearly falls to the ground after being thrown out the window. However, the horizontal motion of the book is independent of its vertical motion, so we can consider the horizontal and vertical components of motion separately. In this case, the initial horizontal velocity of the book allows it to travel a horizontal distance of 31.8 m before landing on the ground.
The window is approximately 31.8 meters above the ground.
Explanation:
To determine the height of the window above the ground, we can use the principles of projectile motion. Since the book is thrown horizontally, its initial vertical velocity is 0 m/s. We can use the following equations:
1. Horizontal motion equation: distance = velocity × time
In horizontal motion, the initial vertical velocity is 0 m/s, so the time taken for the book to land is the same as the time taken to reach the ground.
1. Horizontal motion equation: distance = velocity × time
31.8 m = 12.5 m/s × time
Solving for time:
time = 31.8 m / 12.5 m/s
time ≈ 2.544 s
Now we can use the vertical motion equation to determine the height of the window:
Answers & Comments
To solve the problem, we can use the equations of motion for a projectile. We know that the book is thrown horizontally, so the initial vertical velocity is zero. We can use the following equations:
d = V₀t + 1/2at²
Vf = V₀ + at
where d is the vertical distance traveled, V₀ is the initial vertical velocity, a is the acceleration due to gravity (-9.8 m/s²), t is the time of flight, and Vf is the final vertical velocity (which is also zero at the highest point of the trajectory).
We can use the second equation to find the time of flight:
Vf = V₀ + at
0 = 0 - 9.8t
t = 0 s
This tells us that the book spends no time in the air at the highest point of its trajectory, since it has no initial vertical velocity. Therefore, the vertical distance traveled is simply the height of the window above the ground:
d = 1/2at²
d = 1/2(-9.8 m/s²)(t²)
d = 1/2(-9.8 m/s²)(0 s)²
d = 0 m
So the window is at a height of 0 meters above the ground. This may seem counterintuitive, since the book clearly falls to the ground after being thrown out the window. However, the horizontal motion of the book is independent of its vertical motion, so we can consider the horizontal and vertical components of motion separately. In this case, the initial horizontal velocity of the book allows it to travel a horizontal distance of 31.8 m before landing on the ground.
Answer:
The window is approximately 31.8 meters above the ground.
Explanation:
To determine the height of the window above the ground, we can use the principles of projectile motion. Since the book is thrown horizontally, its initial vertical velocity is 0 m/s. We can use the following equations:
1. Horizontal motion equation: distance = velocity × time
2. Vertical motion equation: distance = (1/2) × acceleration × time^2
In horizontal motion, the initial vertical velocity is 0 m/s, so the time taken for the book to land is the same as the time taken to reach the ground.
1. Horizontal motion equation: distance = velocity × time
31.8 m = 12.5 m/s × time
Solving for time:
time = 31.8 m / 12.5 m/s
time ≈ 2.544 s
Now we can use the vertical motion equation to determine the height of the window:
2. Vertical motion equation: distance = (1/2) × acceleration × time^2
distance = (1/2) × 9.8 m/s^2 × (2.544 s)^2
Solving for distance:
distance ≈ 31.8 m
Therefore, the window is approximately 31.8 meters above the ground.