Mathematics of Falling: The Parabolic Movement

FROM THE LECTURE SERIES: Understanding the Misconceptions of Science

By Don Lincoln, Ph.D., University of Notre Dame

Parabolic movement in physics can be seen in many daily situations, for example, tossing a ball. There are many factors involved in the motion, namely, gravity, velocity, acceleration, and time. Mathematics puts all these factors in formulas that explain how the movement is formed and how it continues. In general, when an object moves forward and upward or downward at the same time, it has a parabolic movement.

Image of a ball in mid-air to illustrate parabolic movement.
The parabolic movement of a thrown object can be calculated by mathematical formulas. (Image: Sergey Nivens/Shutterstock)

When a baseball player throws a ball, it starts a movement in a coordinate system: an x and a y that are perpendicular to each other. The direction of x is parallel to the ground, and y is vertical. The other factor involved in the movement is gravity denoted by g. Velocity and acceleration also matter. How do these elements come together in formulas of parabolic movement and falling?

Gravity and Falling

The elements involved in falling were briefly pointed out. Gravity, or g, is the downward force that pulls the moving object toward the ground, parallel to the y-axis. It has a value of 9.8 meters per second squared. Weight is also an outcome of gravity, but in the formula of falling, it refers to the acceleration due to gravity. The moving object starts with an initial speed, at an initial location, before it gets the gravity acceleration.

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‘d0’, ‘v0’, and ‘a’

‘d0’ refers to the initial location of the object before it starts moving. The object’s initial velocity is denoted by ‘v0’. When there is velocity, there is also acceleration, whether as a constant or a changing element. Acceleration is denoted by ‘a’. With all these pieces of information at hand, one can determine the ball’s location for all time – or ‘t’. Putting all these elements together creates a simple quadratic equation for distance: t = d0 + vt + 1/2at2.

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Simplifying the Equation of Falling

The x and y directions in physics can be completely independently treated. Thus, the quadratic equation can be written in two versions, one for x and one for y, assuming that acceleration is constant and that t equals zero at the moment the ball is released.

Baseball in the air.
To get a simple equation for throwing a ball, many elements should be assumed constant. (Image: Elaine Willcock/Shutterstock)

Due to the downward g of 9.8m/s, ay at t=0 equals -9.8, and ax equals zero since there is no acceleration due to gravity in the horizontal direction. The equation can be further simplified by assuming that both x0 and y0 equal zero at the moment of throwing. The equations will then turn into x = vxt – 1/2gt2 and y = vyt – 1/2gt2. Velocities and acceleration are constant. Thus, they can be replaced by symbols of constant, for example, A and B. What is the shape that these equations can be used for? A parabola.

This is a transcript from the video series Understanding the Misconceptions of Science. Watch it now, on Wondrium.

The Parabolic Movement

In the actual parabolic movement of the thrown ball, the simplifying assumptions do not apply anymore. The mathematical calculations will be much more complicated, and other factors will also influence the motion. Still, in the case of a ball, the object will go upward and forward until it reaches the peak of the movement. Then, the vertical movement gets to zero, and the ball starts going forward and downward. The forward movement is due to the initial energy of throwing, and the downward movement due to gravity.

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The Fall from the Peak

When the object starts to fall, it starts with zero vertical motion, but gravity causes the object to speed up faster and faster. However, there is air resistance present as well. If the assumed ball is a baseball, the parabola looks almost symmetric. On the other hand, if the ball is a beach ball, the movement will be different, even if the weight of the balls is the same. The beach ball is much bigger and faces higher air resistance; hence, lower distance is made in the second half of the way.

Female golfers holding a golf ball / selective focus
Gravity finally stops the upward movement of the object and drags it downward. (Image: Jamesboy Nuchaikong/Shutterstock)

All these assumptions were made for a baseball, which is a small object designed for throwing. Nevertheless, the same formulas apply for other objects, whether it is a feather or a rocket. Consequently, the formula for falling is the same, but the parabolic movement can sometimes be heavily disturbed by other factors, such as air resistance.

Common Questions about Parabolic Movement

Q: Why do objects move in a parabolic motion?

Through experience, almost everyone knows that when an object is thrown, it will go up (or straight, depending on the angle) for a distance, and then back down. Under rules of physics, the vertical position of the object is affected only by a constant acceleration, and horizontal velocity is also normally constant. Thus, the object will have a parabolic movement.

Q: What does parabolic mean in physics?

When an object is thrown, it moves forward but is constantly affected by gravity as well. Consequently, it will move downward toward gravity’s pull, creating a parabolic movement for the object. The arc and the width of the parabola depend on numerous factors.

Q: What is the two-dimensional motion?

When an object moves simultaneously along the x and y axes, it has a two-dimensional movement. The position of an object in two-dimensional space can be plotted by its (x , y) coordinate. A parabolic movement is a kind of two-dimensional motion since the object moves both upward (or down) and at the same time forward.

Q: What is an example of a two-dimensional motion?

A very common example of two-dimensional motion is when an object is thrown forward: a cannon shot, a baseball tossed, or a volleyball being set. There are other examples as well, but throwing and its parabolic movement are among the most familiar ones.

Keep reading
Einstein’s Equivalence Principle: Expanding Our View on Gravity
The Development and Validation of Quantum Field Theory
The Developments in Quantum Mechanics