Chapter 3 Nonlinear Motion

Scalar quantity a quantity that has magnitude but not direction

Vector quantity a quantity that has both magnitude and direction

Vector an arrow drawn to scale used to represent a vector quantity

These represent equivalent vectors:

Vector quantity - a quantity that has both magnitude and direction Vector - an arrow drawn to scale used to represent a vector quantity Scalar quantity - a quantity that has magnitude but not direction

Examples Speed……….. Velocity…….... Acceleration.. Time…………. Distance…….. Force………… scalar vector scalar vector

Addition of Vectors The sum of two or more vectors is called their resultant. To find the resultant of two vectors that are at angles to each other, we use the tip-to-tail method.

Projectile Motion A projectile is any object that is projected by some means and continues in motion by it own inertia. The velocity of a projectile has a horizontal and vertical component. Each component acts independently of the other.

For the vertical motion the acceleration is 9.8m/s 2 downward. For the horizontal motion there is no acceleration. Projectile Drawing.

Projectile Motion The shape of a projectiles path is a parabola. The same range is obtained from two different projection angles that add up to 90° (complementary angles). Maximum range for a projectile is achieved with a projection angle of 45°.

In the presence of air resistance, the trajectory of a high-speed projectile falls short of a parabolic path. A projectile fired horizontally will hit the ground at same time as an object dropped from rest if they are released at the same height. Demo: Ball projector and dropper Animaton: Physics classroom

Example Questions You are driving along in an open car and throw a ball straight up into the air. Neglect air resistance. (a) Where does the ball land relative to the car? Answer: In the car. *

Example: Projectile Motion An object may move in both the x and y directions simultaneously (i.e. in two dimensions) The form of two dimensional motion we will deal with is called projectile motion We may: »ignore air friction »ignore the rotation of the earth With these assumptions, an object in projectile motion will follow a parabolic path

Notes on Projectile Motion: once released, only gravity pulls on the object, just like in up-and-down motion since gravity pulls on the object downwards: vertical acceleration downwards NO acceleration in horizontal direction

Projectile Motion

Rules of Projectile Motion Introduce coordinate frame: y is up The x- and y-components of motion can be treated independently Velocities (incl. initial velocity) can be broken down into its x- and y-components The x-direction is uniform motion a x = 0 The y-direction is free fall |a y |= g

Some Details About the Rules x-direction –a x = 0 – –x = v xo t This is the only operative equation in the x-direction since there is uniform velocity in that direction

More Details About the Rules y-direction – –take the positive direction as upward –then: free fall problem only then: a y = -g (in general, |a y |= g) –uniformly accelerated motion, so the motion equations all hold

Velocity of the Projectile The velocity of the projectile at any point of its motion is the vector sum of its x and y components at that point

(b) While the ball is still in the air you step on the accelerator. Where does the ball land relative to the car? Answer: Behind the car. (c) What if you stepped on the brake instead? Answer: In front of the car. Demo: Cart and ball launcher

Example Questions You drop a ball from the window of a school bus moving a 10 miles/hour. Neglect air resistance. (a) Where does the ball land relative to your hand? Answer: Directly below your hand. *

(b) What is the shape of the path made by the ball seen by someone outside the bus? Answer: A parabola.

Fast-Moving Projectiles Satellites An earth satellite is simply a projectile that falls around the earth rather that into it. Satellites are not free from gravity.

The speed of the satellite must be great enough to ensure that its falling distance matches the earth's curvature. It takes 1 second for an object in free fall to fall 4.9 meters. Earth's surface 'drops' a vertical distance of 4.9 meters for every 8000 meters along the Earth's surface.

This means that a satellite near the Earth’s surface must travel at 8000 meters/second! …or 18,000 miles per hour. For example: The space shuttle orbits the Earth once every 90 minutes.