A rifle bullet is fired from the top of a cliff at an angle of 30o below the horizontal. The initial velocity of the bullet is 800 m/s. If the cliff is 80 m high, how far does it travel horizontally?
138 or rounded to 140!
To find the horizontal distance traveled by the bullet, we need to determine the time of flight.
The motion of the bullet can be divided into two components: horizontal and vertical.
First, let's find the time taken for the bullet to reach the maximum height. We can use the vertical component of the initial velocity:
v_y = v * sin(θ)
v_y = 800 * sin(30°)
v_y = 400 m/s
Using the equation:
Δy = v_iy * t - 0.5 * g * t^2,
where Δy is the displacement in the vertical direction (which is 80 m), v_iy is the initial vertical velocity, t is the time of flight, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Rearranging the equation, we get:
80 = 400t - 0.5 * 9.8 * t^2
Simplifying this quadratic equation, we get:
-4.9t^2 + 400t - 80 = 0
Solving this equation, we find two possible values for t:
t = 0.203 s or t = 40.6 s
Since time cannot be negative, the bullet takes 0.203 s to reach the maximum height.
Next, let's find the time taken for the bullet to hit the ground. At the maximum height, the vertical component of the velocity becomes zero. Using the equation:
v_fy = v_iy + g * t,
0 = 400 - 9.8t,
we find t = 40.8 s.
Now, we can find the horizontal distance traveled by the bullet using the horizontal component of the initial velocity:
v_x = v * cos(θ)
v_x = 800 * cos(30°)
v_x = 800 * √3/2
v_x = 400√3 m/s
To find the distance, we can multiply the horizontal component of the velocity by the time taken for the bullet to hit the ground:
Distance = v_x * t
Distance = 400√3 * 40.8
Distance ≈ 20816 m
Therefore, the bullet travels approximately 20816 m horizontally.