Projectile Motion on a Rotating Earth

Projectile motion refers to the motion of an object that is launched into the air and moves along a curved path under the influence of gravity. When dealing with projectile motion on the surface of the Earth, we typically neglect the rotation of the Earth as it is minimal compared to the scale of the motion. However, in this scenario, the question specifically mentions the latitude of the place, indicating that we need to consider the rotation of the Earth.

Coriolis Effect

The rotation of the Earth gives rise to the Coriolis effect, which is an apparent force experienced by objects moving in a rotating frame of reference. This effect is responsible for the deflection of objects moving in a straight line relative to an observer on the rotating Earth.

Understanding the Problem

In this problem, a particle is thrown vertically upwards with a velocity of 70 m/s at a place where the latitude is 60 degrees. We need to determine how far from the original position the particle will land, taking into account the Coriolis effect.

Analysis

1. Initial Velocity: The particle is thrown vertically upwards with an initial velocity of 70 m/s. Since the motion is vertical, the initial velocity in the horizontal direction is zero.

2. Acceleration due to Gravity: The particle experiences a constant acceleration due to gravity in the vertical direction. The magnitude of this acceleration is 9.8 m/s^2.

3. Coriolis Force: As the particle moves vertically upwards, it will experience the Coriolis force due to the rotation of the Earth. The Coriolis force acts perpendicular to the velocity of the particle and is given by the equation F_c = 2mω x v, where m is the mass of the particle, ω is the angular velocity of the Earth, and v is the velocity of the particle.

4. Angular Velocity of the Earth: The angular velocity of the Earth can be calculated as ω = 2π/T, where T is the period of rotation of the Earth. The period of rotation is given by T = 24 hours = 86400 seconds.

5. Calculating the Coriolis Force: Since the particle is moving vertically upwards, the Coriolis force will act horizontally. The magnitude of the Coriolis force can be calculated using the equation F_c = 2mωv. However, since the velocity of the particle is zero in the horizontal direction, the Coriolis force will also be zero.

6. Vertical Motion: In the absence of the Coriolis force, the particle will continue to move vertically upwards under the influence of gravity until it reaches its maximum height. At this point, the velocity of the particle will be zero.

7. Descent: After reaching its maximum height, the particle will start descending under the influence of gravity. The descent will follow the same path as the ascent, but in the opposite direction.

8. Final Position: The particle will land at the same horizontal distance from its original position as the distance it traveled during its ascent.

Conclusion

In conclusion, considering the Coriolis effect, the particle will land at the same horizontal distance from its