A bullet is found embedded in the wall of a room 2.3 m above the floor. The bullet entered the wall going upward at an angle of 39.8°. How far from the wall was the bullet fired if the gun was held 1.2 m above the floor?
To find the horizontal distance from the wall where the bullet was fired, we can use trigonometry.
Let's break down the information we have:
1. The height of the room: 2.3 m
2. The height at which the gun was held: 1.2 m
3. The angle at which the bullet entered the wall: 39.8°
First, let's find the vertical distance traveled by the bullet.
Using trigonometry, we can determine:
Vertical distance = Total height of the room - Height at which the gun was held
Vertical distance = 2.3 m - 1.2 m
Vertical distance = 1.1 m
Now, let's find the horizontal distance traveled by the bullet.
Using trigonometry, we can determine:
Horizontal distance = Vertical distance / tan(angle)
Horizontal distance = 1.1 m / tan(39.8°)
Horizontal distance ≈ 1.5 m
Therefore, the bullet was fired approximately 1.5 meters away from the wall.
To find the horizontal distance from the wall where the bullet was fired, we can use the horizontal and vertical components of the bullet's velocity.
First, we need to break down the initial velocity of the bullet into its horizontal and vertical components. The horizontal component of velocity remains constant throughout the bullet's flight, while the vertical component changes due to gravity.
Let's denote:
Vi as the initial velocity of the bullet,
θ as the angle of the bullet's trajectory (39.8°),
Vix as the initial horizontal component of velocity,
Viy as the initial vertical component of velocity,
g as the acceleration due to gravity (approximately 9.8 m/s²), and
d as the horizontal distance from the wall where the bullet was fired.
We can calculate the initial horizontal and vertical components of the velocity using trigonometry:
Vix = Vi * cos(θ)
Viy = Vi * sin(θ)
Given:
Vi = unknown
θ = 39.8°
Vix = unknown
Viy = unknown
g = 9.8 m/s²
Next, let's consider the time of flight of the bullet. The bullet is fired from a height of 1.2 m above the floor and hits the wall 2.3 m above the floor. The difference in vertical displacement is 2.3 m - 1.2 m = 1.1 m.
To find the time of flight, we can use the following equation of motion:
Δy = Viy * t + (1/2) * g * t²
Substituting the values:
1.1 m = Vi * sin(θ) * t - (1/2) * 9.8 m/s² * t²
This is a quadratic equation in terms of t, which can be solved to find the time of flight.
Now, we can use the time of flight to find the horizontal distance traveled by the bullet. This can be calculated using the equation:
d = Vix * t
By substituting the known values of Vix and the time of flight obtained earlier, we can solve for d, the horizontal distance from the wall where the bullet was fired.
Huh? I assume you are just showing an example, since none of the numbers matches. In fact, the distance x is found via
x/(2.3-1.2) = cot 39.8°