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) $0^{\circ }$ c) ${60}^{\circ }$ e) none of the b) ${45}^{\circ }$ d) ${90}^{\circ }$ above

Short answer

Answer: e) None of the angles

Step-by-step solution

Understand and identify relevant parameters

In this problem, we are given the projectile motion of a ball thrown at an angle with respect to the horizontal. The relevant parameters are the velocity components ($v_x$ and $v_y$), and the acceleration components ($a_x$ and $a_y$).

Identify the conditions for parallel vectors

For the velocity and acceleration vectors to be parallel, their horizontal and vertical components must be proportional. That is, the ratio of $v_x$ to $a_x$ equals the ratio of $v_y$ to $a_y$.

Determine the acceleration components

Since the only acceleration acting on the ball is gravity (which acts in the vertical direction), we have $a_x=0$ and $a_y=-g$ (with $g$ being the acceleration due to gravity).

Determine the velocity components

The horizontal and vertical components of the velocity can be expressed in terms of the initial velocity $v_0$ and the launch angle $\theta$: $v_x=v_0\cos \theta$ and $v_y=v_0\sin \theta$.

Equate the ratios of the components

Now, we set up the proportionality condition as follows: $\frac{v_x}{a_x}=\frac{v_y}{a_y}$, which leads to $\frac{v_0\cos \theta }{0}=\frac{v_0\sin \theta }{-g}$.

Analyze the proportionality condition

Since $v_0$ is non-zero and $g$ is also non-zero, we can ignore them in the equation and analyze the ratio of the trigonometric functions $\cos \theta$ and $\sin \theta$. However, since the denominator $a_x=0$, the equation leads to an undefined expression, which means that no angle $\theta$ between $0^{\circ }$ and ${90}^{\circ }$ can fulfill the condition for parallel vectors in this case. So, the answer is: e) none of the angles

Key concepts

Velocity Components

When discussing the motion of a projectile, understanding the velocity components is crucial. A projectile, like a ball thrown in the air, has its velocity broken down into two main components: horizontal ($v_x$) and vertical ($v_y$). The horizontal velocity component is calculated using the formula $v_x=v_0\cos \theta$, where $v_0$ is the initial velocity and $\theta$ is the launch angle.
The vertical velocity component can be expressed as $v_y=v_0\sin \theta$.

• These components allow us to analyze the motion in two dimensions separately.
• The horizontal motion is usually constant as there is no horizontal acceleration.
• The vertical motion is affected by gravity, which makes it change over time.

By understanding these components, one can accurately predict the path and behavior of a projectile in motion.

Acceleration Due to Gravity

Gravity plays a substantial role in the motion of any projectile. The only acceleration acting on the projectile is due to gravity, which affects the vertical component of the velocity.

• This gravitational acceleration is generally denoted by $g$ and has a value of approximately $9.81{\text{ m/s}}^2$ directed downward.
• For any projectile, the acceleration in the horizontal direction is zero.
• Thus, $a_x=0$ and $a_y=-g$ (considering downward as negative).

By acknowledging this, one can deduce how a projectile will slow down as it rises, stop for a moment at its peak, and then accelerate downwards. This understanding of acceleration is critical in predicting the projectile's complete motion path.

Vector Parallelism

In the context of projectile motion, vector parallelism occurs when velocity and acceleration vectors point in exactly the same direction. For this to occur, the ratios of their components must be equal. Mathematically, this is represented by the equation:
$$\frac{v_x}{a_x}=\frac{v_y}{a_y}$$

• Parallelism requires both the horizontal and vertical component ratios to be proportional.
• In projectile motion, since $a_x=0$, the equation for parallelism becomes undefined.
• This means the requirement for these vectors to be parallel with a non-zero initial velocity cannot be fulfilled during the projectile's flight.

Understanding vector parallelism helps in analyzing conditions under which a projectile might behave differently if external forces other than gravity were present.

Trigonometric Functions

Trigonometric functions are instrumental in decomposing the projectile's initial velocity into its horizontal and vertical components. They serve as mathematical tools to relate angles with side lengths in right triangles.

• The cosine function is used to find the horizontal velocity component: $v_x=v_0\cos \theta$.
• The sine function determines the vertical velocity component: $v_y=v_0\sin \theta$.
• These functions provide essential links between the geometry of projectile paths and physical behavior.

Apart from their usage in solving projectile motion problems, trigonometric functions are fundamental in many areas of physics and engineering, providing insight into angles and distances."