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A bullet is fired vertically upward with initial speed wo, at latitude ö. Neglecting air resistance, by what distance will it be displaced horizontally when it returns to the ground?
PsyDAG
answered11 years ago
Explain Bot
answered1 year ago
To determine the horizontal displacement of the bullet when it returns to the ground, we need to consider the effect of the Earth's rotation on its motion. The Earth is rotating about its axis, causing the ground to move underneath any objects in the air.
First, let's break down the motion of the bullet. When the bullet is initially fired vertically upwards, it moves straight up against the force of gravity. At the highest point of its trajectory, it reaches a momentary state of zero velocity, also known as the peak or the maximum height.
During the entire duration of the bullet's motion, it experiences a downward acceleration due to gravity, which causes it to fall. However, while the bullet is in the air, the ground beneath it moves horizontally due to the Earth's rotation.
To calculate the horizontal displacement, we need to consider the resulting time of flight and the horizontal velocity due to the Earth's rotation.
1. Calculating the time of flight:
To find the time it takes for the bullet to return to the ground, we can use the vertical motion equations. The equation for the time of flight (t) is:
t = (2 * wo) / g
where wo is the initial vertical velocity (upward) and g is the acceleration due to gravity (which is approximately 9.8 m/s²).
2. Calculating the horizontal velocity due to Earth's rotation:
Next, we need to determine the horizontal velocity provided by the Earth's rotation. At latitude φ, the horizontal velocity due to Earth's rotation is given by:
v_r = vel_earth * cos(φ)
where vel_earth is the velocity of the Earth's rotation (approximately 465.1 m/s at the equator) and φ is the latitude.
3. Calculating the horizontal displacement:
The horizontal displacement can be calculated using the formula:
d = t * v_r
where d is the horizontal displacement, t is the time of flight, and v_r is the horizontal velocity due to Earth's rotation.
By plugging in the values for wo, g, φ, and vel_earth into the equations mentioned above, you can calculate the horizontal displacement of the bullet when it returns to the ground.
First, let's break down the motion of the bullet. When the bullet is initially fired vertically upwards, it moves straight up against the force of gravity. At the highest point of its trajectory, it reaches a momentary state of zero velocity, also known as the peak or the maximum height.
During the entire duration of the bullet's motion, it experiences a downward acceleration due to gravity, which causes it to fall. However, while the bullet is in the air, the ground beneath it moves horizontally due to the Earth's rotation.
To calculate the horizontal displacement, we need to consider the resulting time of flight and the horizontal velocity due to the Earth's rotation.
1. Calculating the time of flight:
To find the time it takes for the bullet to return to the ground, we can use the vertical motion equations. The equation for the time of flight (t) is:
t = (2 * wo) / g
where wo is the initial vertical velocity (upward) and g is the acceleration due to gravity (which is approximately 9.8 m/s²).
2. Calculating the horizontal velocity due to Earth's rotation:
Next, we need to determine the horizontal velocity provided by the Earth's rotation. At latitude φ, the horizontal velocity due to Earth's rotation is given by:
v_r = vel_earth * cos(φ)
where vel_earth is the velocity of the Earth's rotation (approximately 465.1 m/s at the equator) and φ is the latitude.
3. Calculating the horizontal displacement:
The horizontal displacement can be calculated using the formula:
d = t * v_r
where d is the horizontal displacement, t is the time of flight, and v_r is the horizontal velocity due to Earth's rotation.
By plugging in the values for wo, g, φ, and vel_earth into the equations mentioned above, you can calculate the horizontal displacement of the bullet when it returns to the ground.
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