Question

Initial angle of a projectile is \(\theta\) and its initial velocity is \(\mathrm{V}_{0}\). Find the angle of velocity with horizontal line at time \(\mathrm{t}\). (A) \(\sin ^{-1}\left[1-\left\\{g /\left(\mathrm{V}_{0} \cos \theta\right)\right\\} \mathrm{t}\right]\) (B) \(\tan ^{-1}\left[1-\left\\{g /\left(\mathrm{V}_{0} \cos \theta\right)\right\\} \mathrm{t}\right]\) (C) \(\tan ^{-1}\left[\tan \theta-\left\\{g /\left(\mathrm{V}_{0} \cos \theta\right)\right\\} \mathrm{t}\right]\) (D) \(\sin ^{-1}\left[\tan \theta-\left\\{g /\left(\mathrm{V}_{0} \cos \theta\right)\right\\} \mathrm{t}\right]\)

Short answer

The short answer is: \(\alpha = \tan^{-1}\left[\tan\theta-\left\{g /\left(V_{0} \cos\theta\right)\right\}t\right]\)

Step-by-step solution

Break down the initial velocity into horizontal and vertical components

The initial horizontal velocity (\(V_{0x}\)) and initial vertical velocity (\(V_{0y}\)) can be found by using trigonometry: \(V_{0x} = V_0 \cos\theta\) \(V_{0y} = V_0 \sin\theta\)

Find the horizontal and vertical velocities at time t

The horizontal velocity does not change with time, as there is no horizontal acceleration, so: \(V_{x} = V_0 \cos\theta\) For the vertical velocity, the acceleration due to gravity acts against the motion. Thus, we can use the following equation of motion to find the vertical velocity at time t: \(V_{y} = V_{0y} - gt\) Substituting the initial vertical velocity from Step 1, we get: \(V_{y} = V_0 \sin\theta - gt\)

Find the angle of velocity with the horizontal line

The angle of velocity with the horizontal line at time t can be found using the arctangent function: \(\alpha = \tan^{-1}\frac{V_y}{V_x}\) Substituting the horizontal and vertical velocities we found in Step 2, we get: \(\alpha = \tan^{-1}\frac{V_0 \sin\theta - gt}{V_0 \cos\theta}\) Now, we modify the equation by dividing both the numerator and the denominator by \(V_0 \cos\theta\): \(\alpha = \tan^{-1}\left[\frac{\sin\theta - \left(\frac{g}{V_0 \cos\theta}\right)t}{1}\right]\) \(\alpha = \tan^{-1}\left[\sin\theta - \left\{\frac{g}{\left(V_{0} \cos\theta\right)}\right\}t\right]\) Comparing this expression with the given options, we find that it matches option (C). Therefore, the angle of velocity with the horizontal line at time t is given by: \(\alpha = \tan^{-1}\left[\tan\theta-\left\{g /\left(V_{0} \cos\theta\right)\right\}t\right]\)

Key concepts

Angle of Projection

The angle of projection is the angle at which a projectile is launched relative to the horizontal axis. This angle, often denoted by \( \theta \), significantly affects the trajectory of the projectile. It determines how far or how high the projectile will travel. For instance, a higher angle may result in a taller but shorter path, while a lower angle may extend the distance traveled horizontally, but decrease the maximum height.
Understanding the angle of projection is crucial in physics problems related to projectile motion because it impacts other parameters like time of flight, maximum height, and range. By analyzing different angles, one can optimize a projectile's path for specific conditions or requirements.

• An angle of 45° is often considered ideal for achieving maximum range in the absence of air resistance.
• The angle impacts how gravity affects the motion, pulling the projectile down as it travels.

Initial Velocity

Initial velocity, denoted as \( V_0 \), refers to the speed at which a projectile is launched. It's a vector quantity, meaning it has both magnitude and direction. The direction is defined by the angle of projection, \( \theta \), while the magnitude indicates how strong the projectile is launched.
The initial velocity plays a definitive role in determining the projectile's path. A higher initial velocity generally means the projectile will travel further and achieve a larger height. However, this also depends on the angle at which it is launched.

• It is the initial force behind the motion, overcoming gravity and air resistance, setting the projectile in motion.
• The optimal initial velocity is essential to ensure the projectile reaches the desired target or distance.

Horizontal and Vertical Components

The initial velocity of a projectile can be broken down into two components: horizontal and vertical. This separation simplifies the analysis of projectile motion. The horizontal component, \( V_{0x} \), represents the motion along the x-axis and is affected by the cosine of the angle of projection: \( V_{0x} = V_0 \cos\theta \).
The vertical component, \( V_{0y} \), represents the motion along the y-axis and is determined by the sine of the angle: \( V_{0y} = V_0 \sin\theta \). These components are vital since:

• The horizontal component remains constant throughout the projectile's flight due to the lack of horizontal forces acting on it, assuming air resistance is negligible.
• The vertical component changes over time due to gravity. It starts at \( V_0 \sin\theta \) and diminishes until it reaches zero at the peak of the projectile's path and then becomes negative as the projectile descends back to the ground.

The separation into components allows for the use of trigonometry and kinematic equations to solve complex projectile motion problems easily.

Equations of Motion

In the context of projectile motion, equations of motion play a critical role in predicting and understanding the path of a projectile. These equations are derived from Newton's laws of motion and are used to calculate various parameters like displacement, velocity, and time.
For horizontal motion, the equation \( V_x = V_0 \cos\theta \) holds, indicating that horizontal velocity remains unchanged (in the absence of air resistance). In contrast, the vertical motion is influenced by gravity, described by the equation \( V_y = V_0 \sin\theta - gt \), where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, m/s^2 \)).
Using these equations, one can find the angle of velocity with the horizontal as:
\[ \alpha = \tan^{-1} \left( \frac{V_y}{V_x} \right) \]
This specific equation illustrates how to determine the direction of velocity at any time \( t \) during the projectile's flight. By rearranging and substituting known values, problems involving projectile motion become much more approachable and solvable. These equations are an essential toolkit for analyzing and predicting projectile trajectories.