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Projectile motion

What is projectile motion?What is ballistics?

Projectile motion is motion experienced by an object that is thrown near the Earth’s surface: it moves along a curved path under the action of gravity only.

For example – A skateboarder jumps, and then comes down – a firework explodes, and the many glowing pieces fall.

Each of those pieces follows projectile motion.

 

More examples – look at the acrobat being pushed – his motion is parabolic. He’s following a ballistic trajectory.

Same for the soccer ball kicked by this person. These motions are parabolic. They are following a ballistic trajectory.

 

Other examples? Path of a ballistic missile after being launched.

For now, we are leaving out air resistance.

When you kick, throw, shoot or launch an object, once its off and moving, the path of the resulting motion is a parabola.

Ballistic Parabola Projectile Motion minimal GIF

GIF InclinedThrow, by AllenMcC, Wikipedia, CC BY-SA 3.0

The study of such motion is called ballistics.

Such a trajectory is a ballistic trajectory.

Analysis of forces

The only force acting on the object is gravity. It acts downward, thus accelerating the object downward.

What forces keep it moving sideways? None! No horizontal force is needed to maintain the horizontal velocity component of the object.

It keeps moving sideways simply due to inertia (see Newton’s laws of motion.)

 

Simplifying assumptions

The effects of air resistance are negligible

The height of the thrown object is small compared to the size of the Earth

For objects that move fast, and cover long distances, or both, the effects of air resistance will become significant, and we then have to modify our analysis to take this in to account.

For objects that are projected to a great altitude, the gravity that they experience will become less and less as the object gets higher.  But our analysis assumes that the acceleration of gravity is constant.  So in cases where the accel of gravity differs we have to modify our analysis. And get this – we literally can’t do this with most traditional math – we cant use algebra. In order to do this kind of mathematical analysis we need to use calculus!

Further reading

See College Physics, OpenStax at Rice University  or from Physics, LibreTexts

Projectile Motion, College Physics, Open Stax

Examples

Let’s fire a cannonball off of a cliff.

Look at the horizontal and vertical components of velocity as time goes by. What does this tell us?

The horizontal (sideways) component of motion is always independent of the vertical component of the motion.

Now consider this situation:

unknown source

unknown source

PowerPoint presentation on Projectile motion, Conceptual Physics

 

Apps and interactives

Projectile Motion app: PhET

 

Practical Problem solving

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Boy on a small hill drops from a tree projectile motion Giancoli
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Projectile Motion is parabolic assuming constant accel of gravity

Also

Giancoli Physics, Chapter 3

Giancoli Physics, Chapter 3

 

Equations used in projectile motion

(Honors and AP Physics)

Giancoli Physics, Chap 3

Giancoli Physics, Chap 3

Diving off a cliff Part I Projectile Motion

Diving off a cliff Part II Projectile Motion

Diving off a cliff Part III Projectile Motion

Learning Standards

Massachusetts Science Curriculum Framework 2016

1. Motion and Forces: Central Concept: Newton’s laws of motion and gravitation describe and predict the motion of most objects.
1.1 Compare and contrast vector quantities (e.g., displacement, velocity, acceleration force, linear momentum) and scalar quantities (e.g., distance, speed, energy, mass, work).

Common Core Standards for Mathematics – Grades 9-12
Standards for Mathematical Practice:

  • Reason abstractly and quantitatively
  • Model with mathematics
  • Look for and express regularity in repeated reasoning
Number and Quantity – Vector and Matrix Quantities
  • N-VM.1     Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments and use appropriate symbols for vectors and their magnitudes.
  • N-VM.2     Find the components of a vector.
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