Two-dimensional Motion

Chapter 3, Sections 3 and 4

Projectile Motion

• Projectile is object launched into flight whose motion continues by its own inertia

• Trajectory: path of projectile is parabolic• Air resistance is assumed to be negligible• Projectile is pulled by gravity as any object in

free fall

Inertia

• The tendency of an object to continue in its present state of motion

• If an object is moving it will continue moving with constant velocity unless acted upon by a force

• There is no force acting on the projectile in the horizontal direction so the horizontal velocity component remains constant

Range of the Projectile

• Range: Horizontal distance of flight• Range depends on launch angle and velocity • Maximum range obtained from 450 angle• Same range results from any two angles that add up to 900

Satellite Motion

• If launch velocity is enough so projectile path matches earth’s curvature, it becomes satellite and orbits earth

• Earth’s curvature drops 5m vertically for 8000m horizontally

• To reach orbit, satellite must have minimum horizontal velocity of 8 km/s

Projectiles as Satellites

Too slow for orbit Faster but still too slow

8 km/s: circular orbit Faster than 8km/s: elliptical orbit

Projectile Motion Components

• Motion of projectile can be separated into horizontal and vertical components

• Horizontal and vertical motions are independent, not affecting each other

• Time of flight is the same in both • Constant vertical acceleration from gravity• No horizontal acceleration, so horizontal

component of velocity is constant

Vertical and Horizontal Motion

Horizontal Launch

Upward Angle Launch

Solving Projectile Problems

• Separate the problem into vertical and horizontal sections and work separately.

• Resolve velocity into vertical (y) and horizontal (x) components.

• vx = v0cosq• vy = v0sinq

q is angle the velocity vector makes with horizontal (x axis)

Solving Projectile Problems

• Use vertical velocity component only with other vertical quantities in the free fall part of the problem

• Use horizontal velocity component only with other horizontal quantities in the constant velocity part of the problem

Solving Projectile Problems• Solve for the time of flight in one

dimension and use this value to solve for the unknown in the other dimension

• Use constant acceleration (free fall) equations for vertical problem, constant velocity for horizontal.

• Use initial velocity and angle to find x and y components only, never in other parts of the problem.

Airplane and Package (bomb)

Monkey and Zookeeper (Hunter)

Frames of Reference

• Observers using different frames of reference may measure motion differently

• When one reference frame is moving with respect to another, velocities and displacements will be measured differently

• Example: passengers on a moving train and observers at the station observe motion on the train

Relative Velocity

• Relative velocity of one moving object to another is the difference between their velocities relative to some common reference point

• Since velocity is a vector, vector methods must be used to find vector sums and differences

Airplane and the Wind

• Airplane’s velocity vector must be added to wind velocity vector to find velocity with respect to earth

• Tailwind is in the same direction as the plane; headwind is in the opposite direction

Boat on the River