Question

A ball is thrown at an angle between \(0^{\circ}\) and \(90^{\circ}\) with respect to the horizontal. Its velocity and acceleration vectors are parallel to each other at a launch angle of a) \(0^{\circ}\). b) \(45^{\circ}\) c) \(60^{\circ}\) d) \(90^{\circ}\) e) none of the above.

Short answer

Answer: d) 90°

Step-by-step solution

Option a) 0°

When the ball is thrown at an angle of 0°, its velocity vector is completely horizontal. The acceleration vector is always vertical and pointing downwards, so the vectors are not parallel.

Option b) 45°

When the ball is thrown at an angle of 45°, its velocity vector has both horizontal and vertical components. However, the horizontal component of acceleration is always zero, and once again, the vectors are not parallel.

Option c) 60°

When the ball is thrown at an angle of 60°, its velocity vector has both horizontal and vertical components. As always, the horizontal component of acceleration is zero, and the vectors are not parallel.

Option d) 90°

When the ball is thrown at an angle of 90°, its velocity vector is completely vertical. The acceleration vector is also vertical and pointing downwards (due to gravity). Since they are both in the same direction (vertical), the vectors are parallel.

Option e) None of the Above

We have found that the vectors are parallel at 90°, so this option is incorrect.

Final Answer

So, the launch angle at which the velocity and acceleration vectors are parallel to each other is d) 90°.

Key concepts

Vector Analysis

Vector analysis is crucial in understanding the movement of objects in physics, including the motion of projectiles. In the context of projectile motion, vectors represent quantities that have both magnitude and direction, such as velocity and acceleration.

When analyzing the vectors of a projectile, it's important to break them down into their horizontal and vertical components. This is because the only acceleration acting on a projectile is due to gravity, which acts vertically downward. Therefore, while the velocity vector can have both horizontal and vertical parts, the acceleration vector has a vertical component only.

Understanding vector analysis allows us to determine whether velocity and acceleration vectors are parallel. For vectors to be parallel, they need to act along the same line. This condition is met in projectile motion only when a projectile is thrown straight up or straight down, as gravity's influence is aligned with the object's velocity vector.

Kinematics

Kinematics is the study of motion without considering the forces that cause that motion. It uses concepts such as velocity, acceleration, displacement, and time to describe the motion of objects. In projectile motion, kinematics allows us to predict the future position and velocity of an object at any given time.

Using kinematic equations, we can deduce that the horizontal motion of a projectile is constant, as there is no acceleration in the horizontal direction (neglecting air resistance). In contrast, the vertical motion is influenced by gravity, resulting in a vertical acceleration of approximately \( 9.8 m/s^2 \) downward.

Key Kinematic Equations for Projectile Motion

• \[ y = v_{0y}t - \frac{1}{2}gt^2 \]
• \[ v_{y} = v_{0y} - gt \]
• \[ x = v_{0x}t \]

These equations reveal the symmetrical nature of projectile motion and explain why the vectors are only parallel at specific angles of launch, such as \( 90^\circ \).

Angle of Launch

The angle of launch, or the angle at which a projectile is projected into the air, is a determining factor in the shape and distance of the projectile's trajectory. It influences the initial velocity vector's horizontal and vertical components and thus affects how far and how high the projectile will travel.

For an angle of launch at \( 0^\circ \) (horizontal), the maximum range is achieved since all the velocity contributes to horizontal displacement. However, as shown in the textbook problem, at this angle, the velocity and acceleration vectors are not parallel. At \( 90^\circ \) (vertical), the projectile will have maximum height with no horizontal displacement, and this is the scenario where velocity and acceleration are parallel due to gravity acting in the same direction as the motion.

Intermediate angles, like \( 45^\circ \) and \( 60^\circ \) depict a mix of both range and height; however, for parallelism between velocity and acceleration, these angles do not meet the criteria. In summary, the angle of launch significantly affects the dynamics of a projectile's flight, including the relationship between its vectors.