Question

Which of the following shapes is seen in the path of a projectile: straight line, parabola, circle, ellipse, hyperbola?

Short answer

The path of a projectile is a parabola.

Step-by-step solution

Understand Projectile Motion

A projectile is an object thrown into the air with an initial velocity, under the influence of gravity alone. It follows a curved path known as a trajectory.

Identify Motion Characteristics

In projectile motion, the object has a horizontal motion at constant velocity and a vertical motion that is uniformly accelerated. This combination of motions must result in a specific type of path.

Derive Path Equation

The equation governing the motion of the projectile is derived from its horizontal and vertical components: given initial velocity components as \(v_{0x}\) and \(v_{0y}\) the equations of motion are \(x = v_{0x} t\) and \(y = v_{0y} t - \frac{1}{2} g t^2\). Solving these equations together gives the trajectory equation.

Observe the Form of the Equation

By substituting \(t\) from the horizontal equation into the vertical equation, we obtain the trajectory equation \(y = (tan \theta) x - \frac{g}{2 v_{0x}^2} x^2\), which is in the form \(y = ax + bx^2\). This is a quadratic equation.

Determine the Shape

A quadratic equation in the form \(y = ax + bx^2\) describes a parabolic path. Therefore, the path of a projectile is a parabola.

Key concepts

Parabola

When a projectile is launched, it travels in a distinct arc due to the influence of gravity and its initial velocity. This arc is known as a parabola. You might have observed this shape if you've ever thrown a ball or watched a fountain. Parabolas are the result of having a quadratic equation governing their motion, and they have a unique "U" or "arch" shape.
Understanding this shape is important in physics and mathematics because it represents uniform acceleration due to gravity. A parabola is symmetrical, which means it has the same shape on either side of its highest point (vertex).
This symmetry is why objects thrown in the air rise and fall at predictable angles and speeds. Parabolas are not only theoretical; they occur often in nature and engineering. From the paths of water in fountains to the trajectories of planets in space, understanding parabolas helps us make precise predictions and calculations.

Trajectory

The term trajectory is used to describe the path of a projectile. This path is dictated by the combined effects of two components of the projectile's motion - the horizontal motion and the vertical motion.

• Horizontal motion: This is uniform, meaning it remains constant as there is no acceleration acting horizontally (in the absence of air resistance).
• Vertical motion: This is uniformly accelerated due to gravity, which always acts downwards.

These two separate motions combine to create the overall path that we call a trajectory. Imagine watching a soccer ball as it's kicked into the air. The ball follows a curved path or trajectory until it hits the ground. This curve forms the shape of a parabola, showing the impact of both gravity and initial force applied to the ball.
Understanding trajectory is crucial in many fields such as sports, physics, engineering, and even space travel, where precise calculations of projectile paths can determine the success or failure of a mission.

Quadratic Equation

A quadratic equation is a mathematical expression with one variable that is squared, forming an "ax^2 + bx + c = 0" format. In projectile motion, the equation that describes the trajectory is quadratic, which is why the path is parabolic.
The general form of a quadratic equation captures the non-linear relationship between the variables of motion. When simplified, the quadratic formula helps determine the exact path and key points of the projectile's flight, such as its maximum height and range.
For students delving into projectile motion, mastering the quadratic equation is pivotal. It helps in deriving the vertical path (height) in relation to the horizontal path (distance), where the coefficient accompanying the squared term helps shape the curvature of the parabola. Quadratic equations are not only essential for solving projectile motion problems but also appear in diverse areas such as economics and engineering.