Final answer:
The horizontal distance a bullet lands when shot at a 30-degree angle with an initial velocity of 120 m/s is calculated using projectile motion equations. The process involves finding the initial velocity components and the time of flight, then calculating the range with the expression R = v_x × t.
Step-by-step explanation:
To find out how far horizontally from the gun the bullet lands, given it is shot with an initial velocity of 120 m/s at a 30-degree angle to the horizontal, we can use the equations of projectile motion. The horizontal distance, known as the range, can be calculated using the formula:
R = (v^2 × sin(2θ)) / g
where v is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity (approximately 9.81 m/s^2).
First, we calculate the initial horizontal and vertical components of the velocity. The initial horizontal velocity (v_x) is v × cos(θ) and the initial vertical velocity (v_y) is v × sin(θ).
For the given problem:
- v_x = 120 m/s × cos(30°) = 120 m/s × (√3/2)
- v_y = 120 m/s × sin(30°) = 120 m/s × 0.5
Next, we find the time of flight by dividing the vertical component by the acceleration due to gravity and then multiplying by 2. The formula for time of flight is:
t = 2 × (v_y) / g
Once we have the time of flight, we can calculate the horizontal range with:
R = v_x × t
Lastly, by plugging in the values, we can find the horizontal distance the bullet travels.