A bullet is fired with an initial speed of 200 m/s. If a target is 500ft away, at what angle should the rifle be raised in order to hit the target?
To find the angle at which the rifle should be raised to hit the target, we can use the equations of motion and kinematics.
Step 1: Convert the distance to meters.
We know that 1 foot is equal to 0.3048 meters. Therefore, the distance of 500 feet can be converted to meters by multiplying 500 by 0.3048:
Distance = 500 ft * 0.3048 m/ft = 152.4 meters.
Step 2: Analyze the horizontal and vertical components of the bullet's motion.
The initial horizontal velocity of the bullet remains constant throughout its flight, as there are no horizontal forces acting on it. Therefore, the horizontal component of the bullet's velocity is unaffected by gravity and remains 200 m/s.
The initial vertical velocity of the bullet is zero, as it is fired horizontally. The only force acting on the bullet in the vertical direction is gravity, which causes it to accelerate downward.
Step 3: Determine the time of flight.
The bullet will take some time to reach the target. The time of flight can be calculated using the equation:
Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2.
Since the initial vertical velocity (V0y) is zero, the equation simplifies to:
Distance = (1/2) * g * Time^2,
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the values:
152.4 m = (1/2) * 9.8 m/s^2 * Time^2.
Simplifying further, we get:
152.4 m = 4.9 m/s^2 * Time^2.
Solving for Time, we find:
Time^2 = (152.4 m) / (4.9 m/s^2) = 31.2 s^2,
Time = sqrt(31.2) = 5.58 seconds (approx).
Step 4: Calculate the required angle.
Using the formula for horizontal motion, we can find the angle at which the rifle should be raised. The formula is given as:
Horizontal Velocity = Initial Velocity * cos(angle).
Rearranging the equation:
cos(angle) = Horizontal Velocity / Initial Velocity,
cos(angle) = 200 m/s / 200 m/s = 1.
To find the angle, we need to take the inverse cosine (cos^-1) of 1:
angle = cos^-1(1) = 0 radians.
Therefore, the rifle needs to be raised at an angle of 0 degrees (horizontal) in order to hit the target, assuming there are no other horizontal obstacles in the path of the bullet.