Projectile Motion

1. Understand the Given Data:

We are given:

• The maximum range of the projectile, ,
• The acceleration due to gravity, .

The goal is to find the muzzle velocity of the shell.

2. Use the Formula for Maximum Range of a Projectile:

The maximum range of a projectile is given by the formula:

$ R v_0 g v_0 v_0 R = 16000 \, \text{m} g = 10 \, \text{m/s}^2 v_0 160000 400 \, \text{m/s} $.

1. Understanding Projectile motion

In projectile motion, the acceleration is constant and directed downwards due to gravity.
Velocity changes in both magnitude and direction throughout the trajectory.
The angle between two vectors is minimum when they are in the same direction (or as close as possible).

2. Analyzing the Trajectory:

• At the initial point, the angle between velocity and acceleration will be obtuse because the initial velocity has an upward component while the acceleration is downward.
• At the highest point of the trajectory, the vertical component of velocity is zero, and the velocity is purely horizontal. The acceleration, due to gravity, is still acting downwards. Therefore, the angle between velocity and acceleration is exactly 90 degrees.
• Throughout the path, except at the highest point, the angle between the velocity and acceleration vectors is acute, but there is only one point where the angle is both minimum and acute.

3. Minimum and Acute:

At the beginning of the trajectory, the angle is large (greater than 90 degrees).
As the projectile moves towards the highest point, the angle decreases and becomes 90 degrees at the highest point.
After the highest point, the angle decreases further to become an acute angle, and it reaches its minimum value at a single point where the velocity vector is closest to being aligned with the acceleration vector.
Therefore, the angle between velocity and acceleration is both acute and minimum at only one point in the trajectory.

4. Final Answer:

The correct answer is: (A) only one point.

1. Understand the Concept:

A projectile follows a curved path under the influence of gravity. The nature of the trajectory depends on the initial velocity and the absence or presence of air resistance.

2. What Happens in Ideal Conditions (Absence of Air Resistance)?

In the absence of air resistance, the trajectory of a projectile is influenced only by the gravitational force acting vertically downwards. The motion can be broken into horizontal and vertical components:

• The horizontal velocity remains constant, and the horizontal displacement increases uniformly.
• The vertical velocity changes due to gravity, leading to an upward and then downward path.

When we combine these motions, the result is a parabolic trajectory.

3. Examine Each Option:

• (A) Semicircle: A semicircle is not a correct option. This shape would be true only in specific cases, such as when the projectile is launched straight upward and falls back down.
• (B) Ellipse: An ellipse is the path followed by an object in orbital motion under the influence of gravity (like planets around the Sun), not a typical projectile motion.
• (C) A Parabola Always: This is incorrect because, in the absence of air resistance, the trajectory is always a parabola, but the term "always" doesn't make sense in other contexts like circular or elliptical motion.
• (D) A Parabola in the Absence of Air Resistance: This is the correct statement. The trajectory of a projectile is a parabola when air resistance is negligible.

4. Final Answer:
The trajectory of projectile is A Parabola in the Absence of Air Resistance

For the object projected at an angle :

• Initial vertical velocity
• Initial horizontal velocity

For the object projected at an angle :

• Initial vertical velocity
• Initial horizontal velocity

We can calculate the maximum vertical height reached by each object using the vertical displacement equation:

For the object projected at angle :

For the object projected at angle :

Now, to find the ratio of their maximum vertical heights, we divide by :

Therefore, the ratio of their maximum vertical heights is: (option C).