Projectile Motion

Projectile motion refers to the curved path followed by an object that is launched into the air and is subject only to the force of gravity and air resistance (if present). The motion can be analyzed by breaking it into horizontal and vertical components.

Centripetal Acceleration

Centripetal acceleration is the acceleration experienced by an object moving in a circular path. It is always directed towards the center of the circle and its magnitude depends on the object's speed and the radius of the circle.

Given Data
- Initial speed (u) = 20 m/s
- Launch angle (θ) = 30 degrees

Analysis

To find the centripetal acceleration one second after projection, we need to determine the velocity of the projectile at that time.

Horizontal Component of Velocity
The horizontal component of velocity remains constant throughout the motion. It can be calculated using the initial speed and launch angle.

Horizontal velocity (Vx) = u * cos(θ)
= 20 * cos(30)
= 20 * (√3/2)
= 10√3 m/s

Vertical Component of Velocity
The vertical component of velocity changes due to the constant acceleration of gravity. It can be calculated using the initial speed and launch angle.

Vertical velocity (Vy) = u * sin(θ)
= 20 * sin(30)
= 20 * (1/2)
= 10 m/s

Time of Flight
The time of flight is the total time taken by the projectile to return to the same horizontal level from which it was launched. It can be determined using the vertical component of velocity and the acceleration due to gravity.

Time of flight (T) = 2 * Vy / g
= 2 * 10 / 9.8
= 20 / 9.8
≈ 2.04 seconds

Centripetal Acceleration
One second after projection, the projectile will still be in motion. The centripetal acceleration can be calculated using the horizontal component of velocity and the radius of the circular path.

Centripetal acceleration (a) = Vx^2 / r

To determine the radius, we need to find the horizontal distance covered by the projectile in one second. Since the horizontal velocity is constant, the horizontal distance can be calculated as:

Horizontal distance (d) = Vx * t
= 10√3 * 1
= 10√3 m

Since the projectile is launched from the ground, the radius of the circular path is equal to the horizontal distance.

Radius (r) = d
= 10√3 m

Now, we can calculate the centripetal acceleration:

Centripetal acceleration (a) = Vx^2 / r
= (10√3)^2 / (10√3)
= 300 / (10√3)
= 30√3 m/s^2

Therefore, the centripetal acceleration one second after projection is 30√3 m/s^2.