a rifle aimed horizontally and 5 feet above level ground fires a bullet with a muzzle velocity of 3000 ft/sec. How far does the bullet travel from the end of the muzzle before it strikes the ground? (Ignore air resistance)
To calculate how far the bullet travels before it strikes the ground, we can use the basic principles of projectile motion.
First, let's break down the problem into two separate motions: the horizontal motion and the vertical motion. In the horizontal direction, the bullet moves with a constant velocity (excluding any effects of air resistance). In the vertical direction, the bullet experiences the force of gravity, which leads to a downward acceleration.
Since we are only interested in the horizontal displacement, we can ignore the vertical motion for now. The horizontal distance traveled by an object can be calculated using the equation:
Distance = Velocity * Time
In this case, the initial horizontal velocity (muzzle velocity) is 3000 ft/sec, and we need to determine the time it takes for the bullet to strike the ground.
To find the time, we can use the vertical motion equation:
Vertical Displacement = Initial Vertical Velocity * Time + (1/2) * Acceleration * Time^2
In this case, the initial vertical velocity is 0 ft/sec (since the bullet is initially fired horizontally) and the vertical acceleration is the acceleration due to gravity, which is approximately 32 ft/sec^2.
Since the bullet is fired horizontally, it will take the same amount of time to reach the ground as it would if it were dropped from the initial height of 5 feet.
To find the time it takes for the bullet to reach the ground, we can rearrange the vertical motion equation to solve for time:
0 = (1/2) * 32 ft/sec^2 * Time^2 - 5 ft
Now, let's solve for time using this quadratic equation.
Since the coefficient of Time^2 is (1/2) * 32 = 16 ft/sec^2, and the constant term is -5 ft, we can use the quadratic formula:
Time = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 16 ft/sec^2, b = 0, and c = -5 ft. Plugging these values into the quadratic formula:
Time = (-0 ± √(0^2 - 4 * 16 * -5)) / (2 * 16)
Simplifying:
Time = √(80) / 32
Time ≈ 0.631 sec (rounded to three decimal places)
Now that we know the time it takes for the bullet to reach the ground, we can calculate the horizontal distance traveled using the horizontal motion equation:
Distance = Velocity * Time
Distance = 3000 ft/sec * 0.631 sec
Distance ≈ 1893.182 ft
Therefore, the bullet travels approximately 1893.182 feet from the end of the muzzle before it strikes the ground.