A bullet is found embedded in the wall of a room 2.9 m above the floor. The bullet entered the wall going upward at an angle of 36.7°. How far from the wall was the bullet fired if the gun was held 1.1 m above the floor?
x/(2.9-1.1) = cot 36.7°
To solve this problem, we can use the principles of projectile motion. We'll consider the vertical and horizontal components of the bullet's motion separately.
First, let's find the initial vertical velocity of the bullet. We know that the bullet entered the wall going upward at an angle of 36.7°, and the gun was held 1.1 m above the floor. Therefore, the initial vertical velocity can be determined using the equation:
v₀y = v₀ * sin(θ)
where v₀y is the initial vertical velocity, v₀ is the initial velocity of the bullet, and θ is the angle of elevation.
Next, let's find the time taken by the bullet to reach the wall. Since we know the vertical distance covered by the bullet (2.9 m), we can use the equation:
Δy = v₀y * t + (0.5) * g * t²
where Δy is the vertical distance, v₀y is the initial vertical velocity, t is the time taken, and g is the acceleration due to gravity (-9.8 m/s²).
Solving this equation for t, we get:
2.9 = v₀y * t + (0.5) * (-9.8) * t²
Now, let's find the horizontal distance covered by the bullet. We can use the equation:
Δx = v₀ * cos(θ) * t
where Δx is the horizontal distance, v₀ is the initial velocity of the bullet, θ is the angle of elevation, and t is the time taken.
Finally, by substituting the values we've found, we can solve for Δx, which will give us the horizontal distance from the wall where the bullet was fired.
Let's calculate the vertical and horizontal components of the bullet's motion:
v₀y = v₀ * sin(θ)
= v₀ * sin(36.7°)
t = time taken to reach the wall
2.9 = v₀y * t + (0.5) * (-9.8) * t²
Δx = v₀ * cos(θ) * t
= v₀ * cos(36.7°) * t
Now we have two equations with two unknowns (v₀ and t). By solving these equations simultaneously, we can find the values of v₀ and t, and then substitute them into the equation for Δx to determine the horizontal distance from the wall where the bullet was fired.