To determine the time at which the ball's velocity becomes perpendicular to the velocity of projection, we need to analyze the vertical and horizontal components of the ball's motion separately.
Let's break down the initial velocity of the ball into its vertical and horizontal components:
Vertical component: V_y = V_initial * sin(angle) Horizontal component: V_x = V_initial * cos(angle)
Given that the initial velocity (V_initial) is 20 m/s and the launch angle is 30°, we can calculate:
V_y = 20 m/s * sin(30°) = 10 m/s V_x = 20 m/s * cos(30°) = 17.32 m/s (approximately)
The vertical component of the ball's velocity will decrease due to gravity, while the horizontal component will remain constant.
The time at which the ball's velocity becomes perpendicular to the velocity of projection occurs when the vertical component of velocity becomes zero.
Let's consider the vertical motion of the ball:
Vertical velocity component: V_y = V_initial_y - g * t
where:
- V_y is the vertical velocity component at any time t,
- V_initial_y is the initial vertical velocity (10 m/s),
- g is the acceleration due to gravity (approximately 9.8 m/s^2),
- t is the time elapsed.
Setting V_y to zero, we can solve for t:
0 = 10 m/s - 9.8 m/s^2 * t
Solving for t:
t = 10 m/s / 9.8 m/s^2 ≈ 1.02 seconds
Therefore, after approximately 1.02 seconds, the vertical component of the ball's velocity will become zero, and its velocity will be perpendicular to the velocity of projection.