rozana,a fisherwomen throws her net in a lake.the net falls in the lake when it is 2 meters above the ground at 5 meters per second at angle of 60 degrees.(assume that the acceleration due to gravity is -9.8m/s^2).find the vector function of the net at the starting position and find maximum height rozana's net can reach and the time it reached that height.
To find the vector function of the net at the starting position, we need to break down the initial velocity into its x and y components.
Given:
Initial height above ground = 2 meters
Initial velocity = 5 meters per second at an angle of 60 degrees
To find the initial x-component and y-component velocities:
Initial x-component velocity (Vx) = Initial velocity (5 m/s) * cos(60°)
Initial y-component velocity (Vy) = Initial velocity (5 m/s) * sin(60°)
Vx = 5 * cos(60°) = 5 * 0.5 = 2.5 m/s
Vy = 5 * sin(60°) = 5 * 0.866 = 4.33 m/s (approximately)
Now, we can find the vector function of the net:
Let's assume the initial position of the net is (0, 0).
The x-component of the position (x(t)) can be found using the kinematic equation:
x(t) = x0 + Vx * t
Given that x0 (the initial x-position) is 0:
x(t) = Vx * t
The y-component of the position (y(t)) can be found using the kinematic equation for vertical motion:
y(t) = y0 + Vy * t - (1/2) * g * t^2
Given that y0 (the initial y-position) is 2 meters and the acceleration due to gravity (g) is -9.8 m/s^2:
y(t) = 2 + Vy * t - (1/2) * 9.8 * t^2
Therefore, the vector function of the net at the starting position is:
r(t) = (Vx * t, 2 + Vy * t - (1/2) * 9.8 * t^2)
To find the maximum height the net can reach and the time it reached that height, we need to find the vertex of the parabolic function y(t).
The vertex of the parabolic function y(t) = -4.9t^2 + 4.33t + 2 can be found using the formula:
t_vertex = -b / (2a)
In our case, a = -4.9 and b = 4.33.
t_vertex = -4.33 / (2 * -4.9) ≈ 0.44 seconds
To find the maximum height (H), substitute the time at the vertex (t_vertex) into the equation for y(t):
H = y(t_vertex) = 2 + Vy * t_vertex - (1/2) * 9.8 * t_vertex^2
H ≈ 2 + 4.33 * 0.44 - (1/2) * 9.8 * 0.44^2
H ≈ 2 + 1.9 - 0.44
H ≈ 3.46 meters
Therefore, the maximum height the net can reach is approximately 3.46 meters, and it reaches that height after approximately 0.44 seconds.