A ball thrown horizontally at 18.3 m/s from the roof of a building lands 36.3 m from the base of the building. How tall is the building?
To find the height of the building, we need to use the vertical motion of the ball. Since the ball is thrown horizontally, we can ignore any horizontal motion and focus only on the vertical motion. We will use the equation of motion for vertical motion:
h = (1/2) * g * t^2
Where:
h is the height of the building
g is the acceleration due to gravity (9.8 m/s^2)
t is the time it takes for the ball to hit the ground
First, we need to find the time taken for the ball to hit the ground. Since the ball is thrown horizontally, its initial vertical velocity is zero. We can use the following equation to find the time of flight:
t = √(2h / g)
Next, we can find the value of t by rearranging the equation:
t = √(2 * h / g)
To find h, we will use another piece of information given in the problem. The ball lands 36.3 m from the base of the building. Since the ball was in the air for the same amount of time horizontally as it was vertically, we can find the time of flight using the horizontal distance and the initial horizontal velocity (which is constant throughout the motion):
t = d / v
Where:
d is the horizontal distance (36.3 m)
v is the horizontal velocity (18.3 m/s)
Now, we solve for t:
t = 36.3 m / 18.3 m/s
t ≈ 1.986 s
Now that we have the value of t, we can substitute it into the equation for h:
h = (1/2) * g * t^2
h = (1/2) * 9.8 m/s^2 * (1.986 s)^2
h ≈ 19.4 m
Therefore, the height of the building is approximately 19.4 meters.
To determine the height of the building, we need to analyze the vertical motion of the ball.
First, let's consider the horizontal motion of the ball. Since the ball is thrown horizontally, its horizontal velocity remains constant throughout its flight. We are given that the ball was thrown at a speed of 18.3 m/s.
Next, we can determine the time it takes for the ball to land by using the time formula for a horizontally-thrown object:
Time = Distance / Horizontal velocity
Time = 36.3 m / 18.3 m/s = 1.98 s (rounded to two decimal places)
Now, let's focus on the vertical motion. We can use the equation for free fall motion:
Vertical distance = Initial vertical velocity * Time + (1/2) * Acceleration due to gravity * Time^2
Since the ball is thrown horizontally, its initial vertical velocity is 0 m/s. The acceleration due to gravity is 9.8 m/s^2. We know the time it takes for the ball to land is 1.98 s.
Vertical distance = 0 * 1.98 + (1/2) * 9.8 * (1.98)^2
Vertical distance = 0 + (1/2) * 9.8 * (1.98)^2
Vertical distance = 9.61 m (rounded to two decimal places)
Therefore, the height of the building is 9.61 meters.