Chapter 5 Projectile Motion and Satellites

Projectile Motion Describe the motion of an object in TWO dimensions Describe the motion of an object in TWO dimensions Keep it simple by considering motion close to the surface of the earth for the time being Keep it simple by considering motion close to the surface of the earth for the time being Neglect air resistance to make it simpler Neglect air resistance to make it simpler

Projectile Motion The ball is in free fall vertically and moves at constant speed horizontally!!!

Projectile Motion

x y

What happens when we add air resistance? What happens when we add air resistance? Adds a new force on the ball Adds a new force on the ball The force is in the opposite direction to the ball’s velocity vector and is proportional to the velocity at relatively low speeds The force is in the opposite direction to the ball’s velocity vector and is proportional to the velocity at relatively low speeds Need calculus to sort out the resulting motion Need calculus to sort out the resulting motion Lowers the angle for maximum range Lowers the angle for maximum range

Projectile Motion The trick to maximum range is just to keep the object off the ground for as long as possible. The trick to maximum range is just to keep the object off the ground for as long as possible. This allows the horizontal motion to be a maximum since x = v x t This allows the horizontal motion to be a maximum since x = v x t Make range longer by going higher for your starting point Make range longer by going higher for your starting point Make range longer by having more velocity Make range longer by having more velocity

Projectile Motion

Satellites Curvature of the earth enters into our calculations Curvature of the earth enters into our calculations If I start 5 m above the surface, it will still be at 5 m after one second if it is moving 8000 m/sec

Satellites Throw at 8000 m/sec This is about 18,000 mph Earth circumference is 25,000 miles Takes 25000/18000 = 1.4 hours = 84 minutes Higher altitude longer

Satellites Force of gravity on bowling ball is at 90 o to velocity, so it doesn’t change the velocity!!! If no air resistance, gravity doesn’t change speed of satellite, only direction!!!

Communications Satellites Farther out you go, the bigger the circumference of the orbit Farther out you go, the bigger the circumference of the orbit It takes longer for the trip It takes longer for the trip Also, gravity weakens by inverse square law the farther out you go Also, gravity weakens by inverse square law the farther out you go Make the distance so that it takes 24 hours for the orbit Make the distance so that it takes 24 hours for the orbit Satellite is stationary in the sky!!! Satellite is stationary in the sky!!!

The Moon Distance is about 240,000 miles Distance is about 240,000 miles Takes 27.3 days to make an orbit Takes 27.3 days to make an orbit

Elliptical Orbits Give the object a speed a bit greater than 8 km/sec and the orbit will be elliptical Give the object a speed a bit greater than 8 km/sec and the orbit will be elliptical

Elliptical Orbits

Sum of distances from foci to point on the ellipse is a constant!!!

Kepler’s Laws Each planet moves in an elliptical orbit with the sun at one focus of the ellipse. Each planet moves in an elliptical orbit with the sun at one focus of the ellipse. The line from the sun to any planet sweeps out equal areas of space in equal time intervals The line from the sun to any planet sweeps out equal areas of space in equal time intervals The squares of the periods of the planets are proportional to the cubes of their average distances from the sun The squares of the periods of the planets are proportional to the cubes of their average distances from the sun

Kepler’s Laws

Energy Conservation

Escape Speed Throw at > 11.2 km/sec and ball escapes the pull of the earth!!!