Lecture 3: 1D Constant Acceleration & Free Fall

Questions of Yesterday 1a) Is it possible to have +/- velocity and ZERO acceleration? a) YES b) NO 1b) Is it possible to have ZERO velocity and +/- acceleration? a) YES b) NO 2) What is the average velocity in this plot? a) v f b) v f /2 c) between 0 and v f /2 d) between v f /2 and v f t (s) v (m/s) vfvf v f /2 0

Constant Acceleration Motion under Constant Acceleration -> very important All objects on Earth under constant acceleration due to gravity all the time v f - v i t f - t i = a = t (s) v (m/s) vfvf vivi tftf titi

Constant Acceleration Motion under Constant Acceleration -> very important All objects on Earth under constant acceleration due to gravity all the time v f - v i t f - t i = a = t i => 0 t f => t v i = v(t = 0) => v 0 v f => v t (s) v (m/s) v f = v v i = v 0 t f = t t i = 0 = v - v 0 t a

Constant Acceleration: Equations for Motion t i => 0 t f => t v i = v(t = 0) => v 0 v f => v = v - v 0 t a How to determine velocity if you know initial velocity, acceleration and time of motion v = v 0 + at Known Quantities = v 0, a, t Unknown Quantities = v,  x

Constant Acceleration: Equations for Motion How to determine displacement if you know initial velocity, final velocity and time of motion t (s) v (m/s) v v0v0 t0 v+v 0 2 t/2 = v + v 0 2 = xtxt  x = 1/2(v + v 0 )t Known Quantities = v 0, v, t Unknown Quantities =  x, a

Constant Acceleration: Equations for Motion How to determine displacement if you know initial velocity, acceleration and time of motion  x = 1/2(v + v 0 )t v = v 0 + at Known Quantities = v 0, a, t Unknown Quantities =  x, v  x = v 0 t + 1/2at 2

Constant Acceleration: Equations for Motion How to determine velocity if you know initial velocity, displacement, and acceleration of motion  x = 1/2(v + v 0 )t Known Quantities = v 0, a,  x Unknown Quantities = v, t v 2 = v a  x = v - v 0 t a

Practice Problem #1 A Cessna aircraft has a lift-off speed of 120 km/h. a) What minimum constant acceleration does the aircraft require if it is to be airborne after a takeoff run of 240 m? b) How long does it take the aircraft to become airborne?

Practice Problem #2 In the Daytona 500 auto race, a Ford Thunderbird and a Mercedes Benz are moving side by side down a straightaway at 71.5 m/s. The driver of the Thunderbird realizes that she must take a pit stop, and she smoothly slows to a stop over a distance of 250 m. She spends 5.00 s in the pit and then accelerates out, reaching her previous speed of 71.5 m/s after a distance of 350 m. At this point, how far has the Thunderbird fallen behind the Mercedes Benz, which has continued at a constant speed?

Freely Falling Objects Every object on Earth is subject to a constant acceleration due to gravity that pulls it towards Earth g = -9.8 m/s Often convenient to use reference frame with the ground at ZERO, +y away from Earth, -y towards Earth g = -9.8 m/s 2

Free Fall: Concepts & Problem Solving What happens to d during free fall? What happens to v? What happens to a? How does each vary with time? Can use constant acceleration equations of motion with a = g = -9.8 m/s v = v 0 + gt  x = v 0 t + 1/2gt 2 v 2 = v g  x

Free Fall: Concepts & Problem Solving If I throw a ball straight up in the air: By how much does the speed decrease with each second while ascending? By how much does the speed increase with each second while descending? How much time is required for rising compared to falling? Does the distance between 1 s intervals increase, decrease, or stay the same while ascending? Does the distance between 1 s intervals increase, decrease, or stay the same while descending?

Free Fall: Problem #1 If I throw a ball straight up in the air: a) What is the velocity of the ball when it reaches its highest point? b) What is the velocity 1 s before reaching the highest point? c) What is the change in its velocity during this 1 s interval? d) What is its velocity 1 s after reaching its highest point? e) What is the change in its velocity during this 1 s interval? f) What is the change in velocity during the 2 s interval? g) What is the acceleration of the ball during c), e), and f)?

Free Fall: Problem #2 A mountain climber stands at the top of a 50.0 m cliff that overhangs a calm pool of water. She throws two stones vertically downward 1.00 s apart and observes that they cause a single splash. The first stone had an initial velocity of m/s. a) How long after release of the first stone did the two stones hit the water? b) What initial velocity must the second stone have had, given they hit the water at the same time? c) What was the velocity of each stone at the instant it hit the water?

Free Fall: Problem #3 A ball is thrown upward from the ground with an initial speed of 25 m/s. At the same instant another ball is dropped from a building 15 m high. After how long will the balls be at the same height?

Questions of the Day 1)A skydiver jumps out of a hovering helicopter and a few seconds later a second skydiver jumps out so they both fall along the same vertical line relative to the helicopter. 1a) Does the difference in their velocities: a) increase b) decrease c) stay the same 1b) What about the vertical distance between them? 2) I drop ball A and it hits the ground at t 1. I throw ball B horizontally (v 0y = 0) and it hits the ground at t 2. Which is correct? a) t 1 < t 2 b) t 1 > t 2 c) t 1 = t 2