Question

A ball is thrown from ground at an angle between \(0^{\circ}\) and \(90^{\circ} .\) Which of the following remain constant: \(x, y, v_{x}, v_{p}\) \(a_{x}, a_{y} ?\)

Short answer

Answer: The constant variables during the motion of the projectile are horizontal velocity (\(v_x\)) and vertical acceleration (\(a_y\)).

Step-by-step solution

Understand the variables given and break the problem into horizontal and vertical motion

We are given several variables for the ball's projectile motion: position (\(x\), \(y\)), horizontal velocity (\(v_{x}\)), magnitude of velocity (\(v_p\)), and acceleration (\(a_x\), \(a_y\)). We'll first examine the horizontal motion and then the vertical motion to determine which variables are constant.

Analyze the horizontal motion

For the horizontal motion of the projectile, we have the position (\(x\)), the horizontal velocity (\(v_{x}\)), and the horizontal acceleration (\(a_{x}\)). There are no forces acting in the horizontal direction once the ball is in the air, so the horizontal acceleration \(a_x\) is equal to 0. Since there are no forces acting horizontally, the horizontal velocity \(v_x\) will also remain constant. The horizontal position (\(x\)) will keep changing as the ball moves through the air, so it's not constant.

Analyze the vertical motion

For the vertical motion of the projectile, we have the position (\(y\)), the vertical acceleration (\(a_{y}\)), and the magnitude of the projectile's velocity (\(v_p\)). The vertical acceleration \(a_y\) is constant due to gravitational force, which is equal to \(-g\), where \(g\) is the acceleration due to gravity (approximately \(9.81 m/s^2\)). The vertical position (\(y\)) changes as the ball moves through the air, so it's not constant. The magnitude of the projectile's velocity (\(v_p\)) also changes due to vertical acceleration acting on the ball, so it's not constant either.

Conclusion

From our analysis of the horizontal and vertical motion of the projectile, we conclude that the following variables remain constant: horizontal velocity (\(v_x\)) and vertical acceleration (\(a_y\)). The other variables: horizontal position (\(x\)), vertical position (\(y\)), and magnitude of the projectile's velocity (\(v_p\)) do not remain constant during the motion of the projectile.

Key concepts

Understanding Horizontal Velocity

When a ball is thrown in the air at an angle, its movement is described by projectile motion. In layman's terms, projectile motion is the flight path of an object that is launched, hurled, or otherwise projected into the air, subject to only the acceleration of gravity. The horizontal velocity, often denoted as \(v_{x}\), is a key component in this analysis.

Horizontal velocity is the speed at which the projectile moves along the horizontal axis. One important characteristic of horizontal velocity in projectile motion is that it remains constant throughout the flight if we ignore air resistance. This is because, after the initial thrust that propels the ball into the air, there are no additional forces in the horizontal direction affecting its speed. Therefore, \(v_x\), which is the speed component parallel to the ground, stays the same from the moment the ball leaves the hand until the moment it lands.

To help visualize this concept, imagine the ball as a car traveling at a steady speed on a perfectly flat road; there are no reasons for it to speed up or slow down, thus its horizontal velocity is unchanging. In our exercise scenario, while the ball undergoes many changes in its flight pattern, \(v_{x}\) is not one of them—this is a constant value, making calculations predictable and manageable.

Vertical Acceleration in Projectile Motion

In contrast to horizontal velocity, vertical acceleration, represented as \(a_{y}\), is a component of projectile motion that changes the vertical speed of the projectile - but not in the way that might seem intuitive initially. The term acceleration often implies an increase in speed; however, in physics, it refers to a change in velocity, which can mean an increase or decrease in speed or a change in the direction of motion.

For projectiles on Earth, vertical acceleration is due to gravity, usually denoted as \(-g\) and valued at approximately \(9.81 m/s^2\). This gravitational pull works consistently on the projectile throughout its flight and is always directed downward, towards the center of the Earth. Consequently, after the initial upward thrust, as the ball ascends, gravity slows it down until it momentarily stops at the peak of its trajectory. Then, as the ball descends, gravity accelerates it back toward the ground. These changes in vertical speed caused by gravity illustrate why \(a_{y}\) is considered a constant force influencing the object's motion—unaltered by the angle of launch or the initial thrust.

Understanding that \(a_{y}\) is invariably \(-g\), and not dependent on the projectile's path, simplifies calculations related to the vertical component of projectile motion. So, for homework and examination purposes, always remember that vertical acceleration remains consistent and is a linchpin for predicting the vertical behavior of any projectile.

Motion Analysis of Projectiles

To fully grasp the intricacies of projectile motion, performing a motion analysis splits the problem into two separate components: horizontal and vertical. Analysis of motion is essentially breaking down the path into a series of steps or stages to understand each part of the movement and the forces acting at each point.

Motion analysis involves tracking the horizontal and vertical positions, velocities, and accelerations of the projectile over time. For the horizontal aspect, we look at \(v_{x}\) and note its constancy. For the vertical, we monitor the changing vertical speeds due to the constant vertical acceleration \(a_{y}\). By separating these two, studying the motion becomes a more straightforward task because we can focus on each dimension's distinct behavior.

Applying Motion Analysis to the Exercise

Referring to the exercise provided, to analyze the projectile's motion, we acknowledge that the distances traveled horizontally \(x\) and vertically \(y\) are variables that change throughout the ball's trajectory. The magnitude of velocity (\(v_p\)) changes as well because it combines the effects of both horizontal and vertical movements. Thus, by understanding the roles and constant nature of \(v_{x}\) and \(a_{y}\), and recognizing that the other variables are subject to change, a student is better equipped to tackle problems related to projectile motion.

Additionally, the use of diagrams along with motion equations can make concepts clearer. Visual aids, like graphs showing the parabolic trajectory with marked constants and variables, enhance understanding and retention. When students are equipped with the knowledge of what remains constant and what does not, they can approach projectile motion problems with greater confidence and accuracy.