Question

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

Short answer

Answer: The constant variables during the motion of the projectile are the horizontal velocity (\(v_{x}\)), horizontal acceleration (\(a_{x}\), which is zero), and vertical acceleration (\(a_{y}\), a constant downwards value due to gravity).

Step-by-step solution

List the given variables and their meaning

The given variables are: - x: horizontal position of the projectile - y: vertical position of the projectile - \(v_{x}\): horizontal component of the projectile's velocity - \(v_{y}\): vertical component of the projectile's velocity - \(a_{x}\): horizontal acceleration of the projectile - \(a_{y}\): vertical acceleration of the projectile

Analyze horizontal motion

Since there is no horizontal force acting on the projectile after it has been thrown (neglecting air resistance), its horizontal acceleration, \(a_{x}\), is zero. Therefore, its horizontal velocity, \(v_{x}\), remains constant throughout the motion. The horizontal position, x, will also change constantly, as the projectile moves along its trajectory.

Analyze vertical motion

The only vertical force acting on the projectile is gravity, which results in a constant downward acceleration, \(a_{y}\). This means that \(a_{y}\) remains constant, but its value is negative (downwards). The vertical velocity, \(v_{y}\), will continuously change due to the constant gravitational acceleration, eventually reversing its direction when the projectile reaches its maximum height. Lastly, the vertical position, y, will also change constantly, first increasing until the ball reaches its peak height, and then decreasing as it falls back down.

Identify constant variables

In summary, the following variables remain constant during the motion of the projectile: - \(v_{x}\): horizontal velocity - \(a_{x}\): horizontal acceleration (zero) - \(a_{y}\): vertical acceleration (constant downwards value due to gravity) The other variables - x, y, and \(v_{y}\) - will change continuously throughout the trajectory of the ball.

Key concepts

Horizontal Velocity

Understanding the horizontal velocity, expressed as \(v_x\), is crucial for analyzing projectile motion. It's the speed at which an object moves along the horizontal axis. Due to the absence of horizontal forces (when air resistance is negligible), the horizontal velocity remains constant after the object is projected.

From the moment the ball leaves the thrower's hand until it hits the ground, its horizontal velocity doesn't change. It's important to note that this concept is counterintuitive, as we often expect moving objects to slow down over time due to friction or other resistances. However, in an ideal projectile motion environment, \(v_x\) is consistent, allowing for predictable motion along the horizontal plane.

Visualizing Horizontal Velocity

To better visualize horizontal velocity, picture an arrow shot from a bow. Regardless of the arrow's vertical climb and descent, its forward speed does not falter until impact. This example encapsulates the idea of constant horizontal velocity in projectile motion.

Vertical Acceleration

Vertical acceleration in projectile motion, denoted as \(a_y\), refers to the acceleration along the vertical axis, influenced by gravity. It's a constant value of approximately \(9.81 \text{m/s}^2\) directed downwards towards the center of the Earth.

This constant acceleration causes the vertical velocity, represented by \(v_y\), to change uniformly over time. As the ball moves upwards, it slows down due to gravity pulling it back; at its peak, the vertical velocity becomes zero for a brief moment before the ball starts descending, accelerating downwards again.

The Role of Gravity

Gravity plays the lead role here and is the sole reason for vertical acceleration in projectile motion. Even if the ball is thrown straight up or at an angle, gravity will always pull it downward, giving us a predictable vertical acceleration that affects the vertical component of the projectile's trajectory.

Trajectory of a Projectile

The trajectory of a projectile is the path it follows through space, and it can be beautifully represented by a parabolic curve in the absence of air resistance. This shape is determined by the initial velocity, the angle of projection, and the forces acting upon the object, mainly gravity.

When the object is projected upwards at an angle, the trajectory combines both vertical and horizontal movements. Initially, the object climbs, reaches a peak, and then descends, all while maintaining a steady horizontal motion. The result is a path that rises and falls in a smooth, continuous curve.

Decomposing the Trajectory

It's useful to decompose the trajectory into its horizontal and vertical components. The horizontal component shows constant motion, while the vertical component exhibits movement under constant acceleration due to gravity. The interplay between these two motions creates the signature parabolic path of projectile motion, exemplified when you toss a ball, kick a football, or watch a diver jump from a cliff into the sea.