Two-dimensional Motion

Projectile Motion and Circular Motion

Projectile Motion

• Projectile is object whose motion continues by its own inertia.
• Follows a parabolic path
• Constant vertical acceleration from gravity
• No horizontal acceleration, so horizontal component of velocity is constant.
• Horizontal and vertical motions are independent, sharing only the time dimension.
• Distance of flight is called the range
• Range depends on launch angle and velocity
• Maximum range obtained from 45° angle
• Same range results from any two angles that add up to 90°
• If launch velocity is enough so projectile path matches earth's curvature, it becomes satellite and orbits earth.

Solving Projectile Problems

• Separate the problem into vertical and horizontal sections and work separately.
• Resolve velocity into vertical and horizontal components.
• Solve one part of problem for the time and use this value to solve for unknown in the other part.
• Use constant acceleration equations for vertical problem, constant velocity for horizontal.

Circular Motion

• Acceleration is always present since direction is always changing.
• Force must be applied to cause acceleration.
• Can be uniform (constant speed) or variable with tangential acceleration

Centripetal Force

• Force directed towards the center of a circle
• Causes centripetal acceleration.
• Can be supplied by tension in a string, friction, gravity, pushing from another object, etc.
• F_c = mv²/r

Centripetal Acceleration

• Acceleration directed toward a central point
• Changes direction of velocity but not magnitude
• Causes circular motion
• a_c = v²/r

Motion in a Vertical Circle

• Effect of gravity must be considered
• Consider ball on string, moving in circle
• At bottom of circle, tension in string is centripetal force + weight of ball
• At top of circle, tension in string is centripetal force - weight of ball
• If F_c < F_W, ball will fall out of circular path

Critical Velocity

• Minimum velocity needed for object to follow circular path.
• depends only on radius of circle and gravity, not on the mass of the object.
• v_min=(rg)^½

Centrifugal Force

• Not an actual force.
• Only exists for observer in accelerating reference frame.
• Makes objects appear to move away from axis of rotation.
• Actually due to inertia of objects which makes them tend to continue straight line motion.

Frame of Reference

• System for describing location of objects
• In inertial reference frames, law of inertia holds true.
• Non-inertial frames are accelerating and appear to violate law of inertia.
• Relativity is study of relationships between events in different reference frames.

Vocabulary

• Projectile
• Range
• Centripetal
• Centrifugal
• Critical Velocity
• Frame of Reference

Summary

• Projectile motion can be analyzed by separating the vertical and horizontal components of motion.
• Vertical motion has constant acceleration, horizontal motion has constant velocity.
• Circular motion is caused by centripetal force and acceleration, towards center of circle.
• If motion is in vertical plane, effect of gravity must be considered.