Projectile Motion

Step 1: Understanding the projectile motion.
In projectile motion, the speed at maximum height is half the initial speed, which means the vertical component of the velocity is zero at the maximum height. The horizontal component of velocity remains constant.
Step 2: Relation between speed and angle of projection.
At maximum height, the vertical component of the velocity becomes zero. The initial velocity can be broken into components: At the maximum height, the vertical velocity becomes zero, so: Step 3: Solving for .
Using the fact that the initial vertical velocity is related to the angle: Thus, . Step 4: Conclusion.
The correct answer is (4) 45°.

Step 1: Use the formula for the horizontal range.
The horizontal range for a projectile is given by the formula: where is the velocity, is the angle of projection, and is the acceleration due to gravity.
Step 2: Analyze the situation.
Since the velocity is the same for both stones and the only difference is the angle of projection, the horizontal range depends on . For , , and for , . Thus, both stones A and B will have the same range.
Step 3: Conclusion.
Thus, both stones A and B will have the same horizontal range, which corresponds to option (C).

Step 1: Analyze the motion of a projectile.
For projectile motion, the maximum vertical distance (height) is determined by the initial velocity and the angle of projection, while the maximum horizontal distance (range) depends on the initial velocity and the angle.
Step 2: Formula for vertical and horizontal distances.
The maximum vertical distance is given by: and the maximum horizontal distance (range) is given by:
Step 3: Find the ratio.
The ratio of maximum vertical to maximum horizontal distances is: Since , the ratio simplifies to .
Step 4: Conclusion.
Thus, the ratio of maximum vertical to maximum horizontal distances depends on , which corresponds to option (D).

Step 1: Understand momentum at different points.
Momentum is given by: where is the mass and is the velocity.
Step 2: Analyze velocity at lowest and highest points.
At the highest point of the trajectory, the vertical component of the velocity is zero, so only the horizontal component remains. The horizontal component of the velocity is . At the lowest point, the velocity is .
Step 3: Calculate the change in momentum.
The change in momentum is the difference between the momentum at the highest point and the lowest point. Since the horizontal component of velocity is constant, the change in momentum only depends on the vertical component:
Step 4: Conclusion.
Thus, the change in momentum is , which corresponds to option (C).