Instructor: Dr. Tatiana Erukhimova Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lectures 5, 6

If a=ac=Const:

A “police car” problem (cont.) x2 – x1 = 3.5 km x1 x2 x=0 V3=20m/s a=0 V1=30m/s V2=40m/s a=const ap=kt V(t=0)=0 You start moving from rest with constant acceleration. There is a police car hiding behind the tree. The policeman has a metric radar. He measures your velocity to be 30 m/s. While the policeman is converting m/s to mph, you continue accelerating. You meet another police car. This policeman measures your velocity to be 40 m/s. You also notice the police, drop your velocity to 20 m/s and start moving with a constant velocity. However, it is too late. This police car starts chasing you with acceleration kt (k is a constant). After some distance he catches you.

A “police car” problem x2 – x1 = 3.5 km x1 x2 x=0 V3=20m/s a=0 a=const ap=kt V(t=0)=0 1. What was your acceleration before you meet the second police car? 2. How long did you travel from x1 to x2? 3. Find x1 4. At which distance does the police car catch you? 5.Convert the velocity from m/s to mph

An object’s velocity is measured to be , where =4.00 m/s and =2.00 m/s3. At t=0 the object is at x=0. Calculate the object’s position and acceleration as functions of time.

Free fall g=9.8 m/s2=32 ft/s2 g-positive! On planet Earth, if you neglect air resistance, any body which is dropped will experience a constant acceleration, called g, independent of its size or weight. g=9.8 m/s2=32 ft/s2 g-positive!

Galileo Galilei (1564-1624), the basic law of motion a = g = const v Galileo Galilei (1564-1624), the basic law of motion a = g = const for all bodies independently on their masses

Galileo's “Law of Falling Bodies” distance (S) is proportional to time (T) squared

Galileo’s notes

Free fall

A ball is dropped vertically down (no air resistance) from height H. Find the position x(t) and velocity v(t) of the ball as a function of time. How long is the ball in the air?

A person throws a ball upward into the air with an initial velocity of 15 m/s. Calculate How much time does it take for the ball to reach the maximum height? How high does it go? How long is the ball in the air before it comes back to the thrower’s hands The velocity of the ball when it returns to the thrower’s hand At what time t the ball passes a point 8.00 m above the person’s hand

A ball is thrown vertically upward with a velocity of magnitude v1 from a window at height H. What is the ball’s position and velocity at any time moment? How long does it take to reach the highest point? How long does it take to reach the ground? What is the velocity of the ball when it hits the ground?

A ball is thrown vertically upward with a velocity of magnitude v1 from a window at height H. In addition to gravity acting on the ball there is another force so that the acceleration in the up direction is –g+t where  is a constant and t is the time. What is the ball’s position when the acceleration is zero?

A rat is running with known constant velocity v1 when she passes you at t=0. You start from rest with acceleration c1 where c1 is an unknown constant. What does c1 have to be in order to catch the rat after she has gone a distance D?

Falling with air resistance

Falling with air resistance When you fall in vacuum, the only force acting on you is gravity mg, and you experience a constant acceleration g independently on your mass, size, or shape. When you fall through the air or any other medium, you will experience the drag force which will eventually reduce your acceleration to zero, so you will fall with a constant velocity called the terminal velocity. This terminal velocity does depend on your mass, size, shape and even the clothing you wear. Before doing any math, we can experimentally find this dependence using such a simple thing as coffee filters.

Terminal Velocity with Coffee Filters where is the resistance force. A penny and a quarter dropped from a ladder land at the same time (air resistance is negligible). A coin dropped in a coffee filter from a ladder lands later than a coin without coffee filter (the terminal velocity is smaller for larger cross-section area). A quarter dropped in a coffee filter will land faster than a penny in a coffee filter (the terminal velocity is larger for larger mass) Two identical coins dropped in coffee filters of different diameters land at different times (the terminal velocity is smaller for larger cross-section area). A parachute jumper reaches the terminal speed almost immediately as soon as the parachute is opened. A coffee filter is an excellent parachute for a small load. It reaches the terminal velocity over a foot of a free fall.

Resistance force: Terminal velocity: A – area of the projectile For a spherical projectile in air at STP: Terminal velocity: A 70-kg man with a parachute: vT ~ 5 m/s A 70-kg man without a parachute: vT ~ 70 m/s

Have a great day! Reading: Chapters 3, 4