A bullet is fired with a velocity of 500m/s at 30 degree north latitude . calculate the consequent horizontal deflection of the bullet as it hits a target 250metre away.also find the vertical displacement of the bullet due to gravity.

To solve the problem of a bullet fired at a velocity of 500 m/s at a 30-degree angle, we need to analyze both the horizontal deflection due to the Coriolis effect and the vertical displacement caused by gravity. Let's break this down step by step.

Understanding the Components of Motion

The bullet's motion can be divided into two components: horizontal and vertical. The horizontal component is influenced by the Coriolis effect, while the vertical component is affected by gravity.

Calculating Horizontal and Vertical Components of Velocity

First, we need to find the horizontal (Vx) and vertical (Vy) components of the bullet's initial velocity using trigonometric functions:

• Vx = V * cos(θ)
• Vy = V * sin(θ)

Where:

• V = 500 m/s (initial velocity)
• θ = 30 degrees

Calculating these components:

• Vx = 500 * cos(30°) = 500 * (√3/2) ≈ 433 m/s
• Vy = 500 * sin(30°) = 500 * (1/2) = 250 m/s

Time of Flight

Next, we need to determine how long the bullet is in the air while traveling horizontally to the target 250 meters away. The time of flight (t) can be calculated using the horizontal distance (d) and the horizontal velocity (Vx):

• t = d / Vx

Substituting the values:

• t = 250 m / 433 m/s ≈ 0.577 seconds

Calculating Vertical Displacement Due to Gravity

Now, we can find the vertical displacement (h) caused by gravity during this time. The formula for vertical displacement under constant acceleration (gravity) is:

• h = Vy * t - (1/2) * g * t²

Where g (acceleration due to gravity) is approximately 9.81 m/s². Plugging in the values:

• h = 250 m/s * 0.577 s - (1/2) * 9.81 m/s² * (0.577 s)²

Calculating each term:

• h = 144.25 m - (0.5 * 9.81 * 0.333) ≈ 144.25 m - 1.63 m

Thus, the vertical displacement is:

• h ≈ 142.62 m

Horizontal Deflection Due to Coriolis Effect

Now, let's consider the horizontal deflection caused by the Coriolis effect. The formula for the Coriolis deflection (dC) is given by:

• dC = 2 * ω * t * Vx

Where ω is the angular velocity of the Earth, approximately 7.27 x 10^-5 rad/s. Substituting the values:

• dC = 2 * (7.27 x 10^-5 rad/s) * (0.577 s) * (433 m/s)

Calculating this gives:

• dC ≈ 0.017 m or 17 mm

Summary of Results

In summary, as the bullet travels 250 meters:

• The vertical displacement due to gravity is approximately 142.62 meters downward.
• The horizontal deflection due to the Coriolis effect is about 0.017 meters (or 17 mm) to the right.

This analysis illustrates how both gravity and the Earth's rotation can significantly influence the trajectory of a projectile. Understanding these effects is crucial in fields such as ballistics and physics.