Lecture 9, by I. Bars

In this lecture we discuss projectile motion and orbiting motion as a consequence of Newton's laws of motion and the universal law of gravity. A few examples of projectile motion are a cannonball fired, baseball thrown, a socccer ball kicked, or an athlete's long jumping. Even fireworks and water fountains are examples of projectile motion. Motion of satellites and planets in orbit is an extension of projectile motion, and they follow Kepler's laws of motion.

Credit for graphics, CD-Library, Hewitt.

• Nonlinear motion.

Motion of a projectile is not along a straight  line. It is the superposition of linear motion along the x-direction and along the y-direction. Examples: a plane in wind, a boat crossing a river, a cannon ball.
 

The motion along x is completely independent than the motion along y.

Galileo was the first to understand this notion. He concluded that all projectiles follow a curve called the parabola.

An object is controlled by two independent motions. So an object projected horizontally  will reach the ground in the same time as an object dropped vertically. No matter how large  the horizontal velocity is, the downward pull of gravity is always the same.

Question: Boy on tower 4.9 m high throws a ball 20 m downrange. What is his pitching speed? (see Next Time Questions)
 

• Projectile. Gravitational acceleration, velocity vectors. The acceleration is constant, the initial velocity is v₀=(v_0x,v_0y), the initial position is x₀=(x₀,y₀)=(0,0). Superposition of horizontal and vertical motion at any instant of time t is constructed as follows:

 

acceleration	ax = 0,	ay = - g = - 9.8 m/s2
velocity	vx(t) = v0x	vy(t) = v0y - gt
position	x(t) = x0 + v0x t	y(t) = y0 + v0y t - gt2/2

 

Projectile motion (e.g. firing a cannon). Choose initial velocity (magnitude and angle) obtain height, horizontal distance, time to go up and down. See also another link for projectile motion with air resistance.

Question: Monkey and Zookeeper. Where should he aim?
 

• Satellites. If the initial speed of a projectile is large enough the projectile can become a satellite. If the speed is just enough the orbit is circular, if it is somewhat larger the orbit is elliptical, if it is too large, the object escapes.
• How large is the launching speed? the escape speed?
• The Earth curves approximately 5 meters downward for every 8000 meters along its horizon

Since a horizontally-launched projectile falls approximately a vertical distance of 5 meters in its first second of motion (really 4.9 m = g/2), an orbitting projectile must be launched with a horizontal launching speed of at least 8 km/s (i.e. about 29,000 km/h, or 18,000 m/h).  More details here.
 
• Te escape speed from Earth is 11.2 km/s. The escape speed from the Sun is 42.5 km/s.

• Orbital Motion. Satellites and  planets in orbits obey  Kepler's laws of planetary motion (see also this).  Kepler (1571-1630) obtained his laws by fitting elliptical curves to his mentor Tycho Brahe's (1546 - 1601) detailed data on positions of planets over 20 years.

• An animated illustration of Kepler's laws is found here.
• Newton combined the law of universal gravity with his three fundamental laws of motion to derive Kepler's third law, thus justifying the inverse square law, i.e. the 1/r² behavior of the gravitational force. He thus explained why orbits of planets are ellipses in agreement with Kepler. This is an exciting achievement - i.e. reaching a complete and fundamental understanding by starting from first principles !!!
• Total energy ( PE + KE ) is conserved during orbital motion. PE increases with distance, KE decreases with distance (similar to throwing an object up in the air). For circular orbits PE is constant (same distance), therefore KE is constant, therefore speed is constant. For elliptic orbits PE is largest at apogee (farthest point on ellipse) and smallest at perigee (nearest point on ellipse), while KE is opposite. Therefore fastest motion is at perigee and slowest motion is at apogee.

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