A sniper fires a bullet at 120 m/s at 30° above the horizontal from the roof top of a 35 m high parking garage. If the bullet strikes the level ground beside the parking garage:
a) How long was the bullet in the air?
b) How far from the base of the parking garage did the bullet land?
c) At what angle did the bullet land?
See previous post: Fri,5-2-14,2:37 AM.
To answer these questions, we can break down the problem into the horizontal and vertical components of motion.
First, let's find the time of flight, which will help us answer the first question, "How long was the bullet in the air?"
a) To find the time, we need to find the time it takes for the bullet to reach the ground. We can use the vertical motion equation:
h = ut + (1/2)gt²
Where:
h = initial vertical displacement (35 m since the bullet was fired from a 35 m high parking garage)
u = initial vertical velocity (u = usinθ, where u = 120 m/s and θ = 30°)
g = acceleration due to gravity (9.8 m/s², assuming no air resistance)
t = time of flight
Plugging in the values:
35 = (120sin30)t + (1/2)(9.8)t²
Simplifying:
35 = 60t + 4.9t²
Rearranging the equation to get a quadratic equation:
4.9t² + 60t - 35 = 0
Solving this quadratic equation, we find two possible values for t, but we will only consider the positive value since time cannot be negative.
The positive value of t (time of flight) will give us our answer.
b) To find the horizontal distance traveled by the bullet (i.e., how far from the base of the parking garage it lands), we can use the horizontal motion equation:
s = ut
Where:
s = horizontal distance traveled
u = initial horizontal velocity (u = ucosθ, where u = 120 m/s and θ = 30°)
t = time of flight
Plugging in the values:
s = (120cos30)t
Now we can substitute the value of t that we found in part a) into this equation to get the horizontal distance.
c) Finally, to find the angle at which the bullet lands, we can use trigonometry. We know the horizontal distance traveled and the vertical distance (which is 35 meters) from the problem statement. We can use the inverse tangent function:
θ = arctan(vertical distance / horizontal distance)
Plugging in the values, we can find the angle at which the bullet lands.