To determine the position and velocity of the cannonball at 5 seconds, we need to analyze its motion using the given initial velocity and angle of inclination.
Let's break down the initial velocity into its horizontal and vertical components. The horizontal component is given by:
Vx = V * cos(theta)
where V is the initial velocity and theta is the angle of inclination. Substituting the given values:
Vx = 141.4 m/s * cos(45°) ≈ 100 m/s
The vertical component is given by:
Vy = V * sin(theta)
Vx = 141.4 m/s * sin(45°) ≈ 100 m/s
Now, we can analyze the motion of the cannonball separately in the horizontal and vertical directions.
Horizontal Motion: Since there is no horizontal acceleration (assuming no air resistance), the horizontal velocity remains constant at 100 m/s. Therefore, the horizontal position can be calculated as:
x = Vx * t
where t is the time. Substituting the given time:
x = 100 m/s * 5 s = 500 m
The horizontal position of the cannonball at 5 seconds is 500 meters.
Vertical Motion: In the vertical direction, we can analyze the motion using the equations of motion under constant acceleration. The acceleration in the vertical direction is due to gravity, which is approximately 9.8 m/s^2.
Using the equation:
y = Vyi * t + (1/2) * a * t^2
where y is the vertical position, Vyi is the initial vertical velocity, t is the time, and a is the acceleration, we can calculate the vertical position at 5 seconds.
Vyi = Vy = 100 m/s (the initial vertical velocity)
y = Vyi * t + (1/2) * a * t^2 y = 100 m/s * 5 s + (1/2) * 9.8 m/s^2 * (5 s)^2 y ≈ 250 m + 122.5 m y ≈ 372.5 m
The vertical position of the cannonball at 5 seconds is approximately 372.5 meters.
The velocity of the cannonball at 5 seconds can be calculated by combining the horizontal and vertical components:
V = sqrt(Vx^2 + Vy^2)
V = sqrt((100 m/s)^2 + (100 m/s)^2) V ≈ sqrt(20000 m^2/s^2) V ≈ 141.4 m/s
The velocity of the cannonball at 5 seconds is approximately 141.4 m/s.
Therefore, at 5 seconds, the cannonball is approximately 500 meters horizontally, 372.5 meters vertically, and has a velocity of 141.4 m/s.