To determine the maximum displacement in the x-direction for a projectile launched from the origin with an initial velocity vector represented as **u = u1i + u2j**, we need to analyze the motion in both the x and y directions separately. The acceleration due to gravity acts downward along the negative y-axis, which will affect the vertical motion but not the horizontal motion directly.
Breaking Down the Motion
The initial velocity can be split into two components:
- Horizontal component (x-direction): u1
- Vertical component (y-direction): u2
In projectile motion, the horizontal displacement is influenced by the horizontal velocity and the time of flight, while the vertical motion is influenced by gravity.
Time of Flight Calculation
To find the time of flight, we need to determine how long the projectile remains in the air. The vertical motion can be described by the equation:
y = u2t - (1/2)gt²
At the maximum height, the vertical velocity becomes zero. The time to reach maximum height (t_peak) can be found using:
0 = u2 - gt_peak
Solving for t_peak gives:
t_peak = u2/g
Since the time to ascend is equal to the time to descend, the total time of flight (T) is:
T = 2 * t_peak = 2u2/g
Horizontal Displacement Calculation
The horizontal displacement (R) can be calculated using the horizontal component of the velocity and the total time of flight:
R = u1 * T
Substituting the expression for T, we get:
R = u1 * (2u2/g) = (2u1u2)/g
Final Result
Thus, the maximum displacement in the x-direction is:
R = (2u1u2)/g
From the options provided, the correct answer is:
1) 2u1u2/g
This analysis shows how the components of initial velocity and the effect of gravity interact to determine the projectile's trajectory and horizontal displacement. Understanding these principles is crucial in physics, especially in kinematics and projectile motion.
