Final answer:
The ball's momentum after 0.2 seconds is calculated by finding its horizontal and vertical velocity components, considering the effect of gravity, and then using these to compute the momentum components. The magnitude of the momentum vector is approximately 6.018 kg*m/s, and the direction is about 25.74 degrees above the horizontal.
Step-by-step explanation:
To determine the momentum of the ball after 0.2 seconds, we must first calculate the components of the momentum and then construct the magnitude and direction from those components. The ball has a mass of 250 g, which we convert to kilograms as 0.250 kg, and it is thrown with an initial velocity of 25 m/s at a 30° angle with the horizontal. We can break down this velocity into horizontal (Vx) and vertical (Vy) components:
Vx = V*cos(θ) = 25 m/s * cos(30°) = 21.65 m/s
Vy = V*sin(θ) = 25 m/s * sin(30°) = 12.5 m/s
Since there's no air resistance, the horizontal component of the velocity (and therefore momentum) does not change over time. However, gravity affects the vertical component. The acceleration due to gravity is 9.8 m/s2 downward, and after 0.2 s the change in vertical velocity (ΔVy) can be found by:
ΔVy = g*t = 9.8 m/s2 * 0.2 s = 1.96 m/s
This change will be subtracted from the initial vertical velocity component to get the vertical component after 0.2 s, since gravity is acting in the opposite direction:
Vy(t) = Vy - ΔVy = 12.5 m/s - 1.96 m/s = 10.54 m/s
The momentum in the vertical and horizontal directions can now be calculated and combined to find the total momentum of the ball:
Px = m*Vx = 0.250 kg * 21.65 m/s = 5.4125 kg*m/s
Py = m*Vy(t) = 0.250 kg * 10.54 m/s = 2.635 kg*m/s
Finally, to find the magnitude of the momentum vector, we use the Pythagorean theorem:
P = √(Px2 + Py2) = √(5.41252 + 2.6352) = √(29.295 + 6.9432) = √(36.2382) = 6.018 kg*m/s
The direction can be found by taking the arctan of the ratio of Py to Px:
θ = arctan(Py/Px) = arctan(2.635/5.4125) = 25.74°
Therefore, the magnitude of the ball's momentum after 0.2 seconds is approximately 6.018 kg*m/s, and the direction is approximately 25.74° above the horizontal.