Physics presentation: projectile motion

There is nothing more awesome than watching physics being applied successfully to the real-world. Make this a meme: APPLIED PHYSICS IS APPLIED. (Movie link: "DC SHOES: HOOPS COMMERCIAL.")

We'll analyze the principles behind this type of motion, and the equations used to analyze this motion.

Some working definitions: a projectile is an object that is subject only to the force of gravity once underway, and we'll consider the simplest (but not necessarily realistic) case where drag is negligible.

With these assumptions, then the trajectory--the path that this projectile travels along--has certain special properties.

Let's see how we can extend our previous understanding of objects moving vertically in free fall to projectile motion, and watch various examples of projectiles in motion.

If we shoot a ball vertically upwards, its upwards and subsequent downwards motion is only subject to the force of gravity (neglecting drag). This can be considered a special case of projectile motion.

Suppose that the cart that vertically launches the ball moves at a constant speed horizontally, here, along a smooth track.

When this horizontally moving cart launches a ball vertically...

...the ball will move along a trajectory...

...such that it will subsequently land back into the cart. This means that the horizontal motion of both ball and cart were always in sync, and that their horizontal motion is independent of the vertical motion of the ball. (Movie link: "110725-1240640-r.")

So projectile motion depends on two independent ingredients--vertical: free fall motion; horizontal: constant velocity motion.

If there is no horizontal motion, then projectile motion is the simple vertical free fall case. Here, stacking two anvils with a generous amount of gunpowder sandwiched between them results in...a vertical anvil trajectory. (Movie link: "Downieville Gold Rush Anvil Launch.")

Which initial velocity component(s) for the anvil (v_0x, v_0y) is/are zero? positive? Negative?
Which (constant) acceleration component(s) for the anvil (a_x, a_y) is/are zero? positive? Negative?

Driving a car off of a cliff results in another example of projectile motion, but remember that this is just vertical free fall, with the constant horizontal motion of the car's initial velocity as it drove off of the cliff. (Movie link: "Car Off Cliff.")

Which initial velocity component(s) for the car (v_0x, v_0y) is/are zero? positive? Negative?
Which (constant) acceleration component(s) for the car (a_x, a_y) is/are zero? positive? Negative?

Not content to drive a car off of a cliff, here a car is launched diagonally upwards, and then hit with an anti-tank rocket (which may or may not be a computer generated special effect). Again this is just vertical free fall, with constant horizontal motion. (Movie link: "Jeremy Clarkson - Hot Metal.")

Which initial velocity component(s) for the car (v_0x, v_0y) is/are zero? positive? Negative?
Which (constant) acceleration component(s) for the car (a_x, a_y) is/are zero? positive? Negative?

Hijinks aside, it's time to look at the boring but important equations that describe projectile motion.

Vertical motion is described with the same set of free fall equations that we have seen before.

With the sometimes necessary quadratic formula...

The only new equation here reflects the constant horizontal motion that goes on simultaneously with the vertical free fall motion. Since horizontal velocity never changes, horizontal acceleration is zero, so the only important equation is how horizontal displacement increases linearly with elapsed time.

So whereas we had five equations to describe vertical free fall motion, we only need one more equation--constant horizontal motion--to fully describe projectile trajectories.

Closing note--trajectory motion is merely vertical free fall, with an added horizontal velocity component: so two billiard balls released simultaneously, with one dropped from rest, the other with an initial horizontal velocity will have the same vertical motion, and must hit the floor at the same time. (Movie link: "Shoot-n-Drop.")