Physics 1110: Motion in 1D Announcements: Wasn t around yesterday; Had a 65 ft spruce tree crash thru my roof in the wind storm Good talk on Sat at 2pm in G1B30 by Professor Emeritus Albert Bartlett CAPA homework due Tuesday at 10pm Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/ 1
Vectors Vectors have magnitude and direction velocity, acceleration, displacement Scalars only have magnitude speed, temperature, mass, volume Vector quantities are indicated by bold and/or an arrow above: Can use arrow to represent direction & magnitude 2
Reference frame To solve any problem, need to pick a reference frame and conduct all calculations in that frame Standard 2D reference frame is motionless with respect to the Earth and has x and y axes as shown For 1D motion, we choose the x-axis to lie along the direction of motion so the y-axis is unnecessary -x +y -y +x 3
1D kinematics (motion) Define two quantities related to change in position: Distance is a scalar which can be thought of as the ground covered by the chosen path Displacement is a vector between the start and end points whose magnitude is the distance between the start and end points: In 1D, can simplify to. The vector direction can be along positive or negative axis so the signs of x 2 and x 1 are still relevant. 4
Displacement and Distance A person swims one lap (roundtrip) in an Olympic-sized pool (50m long). What is the distance traveled and the displacement? Place origin at the starting point Outbound trip: Return trip: Distance = 100 m -100 50-100 50 0 50 100 0 50 100 x (m) x (m) Displacement = Δx = x 2 - x 1 = 0-0 = 0 m 5
Clicker question 1 Set frequency to BA An ant crawling along the floor follows a semi-circular path, going half way around the circumference of a circle of radius R. The distance traveled, and the displacement of the ant, are respectively: A: π R and π R B: 2 R and π R C: π R and 2 R D: π R and zero E: none of these 6
Clicker question 2 Set frequency to BA An object goes from one point in space to another. After it arrives at its destination, the size of its displacement is: A: either greater than or equal to B: always equal to C: either smaller than or equal to D: could be smaller or larger than the distance it traveled. 7
Speed and velocity Speed (scalar) is distance divided by time Velocity (vector) is displacement divided by time: In 1D we drop the vector notation but remember to keep the sign of x 1 and x 2. 8
Clicker question 3 Set frequency to BA A person starts in Boulder, drives to Denver (50 km away) in 1 hour, stays in Denver 1 hour, then speeds back to Boulder in 30 minutes. start 50 km in 1 hour Denver 5inish 50 km in 1/2 hour What is the average speed of the round trip? A: 25 km/hr B: 67 km/hr C: 40 km/hr D: 75 km/hr E: none of these (wait 1 hour) Ave speed = total distance/total time = 100 km / 2.5 hours = 40 km/hr 9
Clicker question 4 Set frequency to BA Q. In 2002, Steve Fosset traveled 32000 km around the world in 13 days (1.0 10 6 s) in a balloon, landing at his started point. What was his average velocity? A. 0.032 m/s B. 32 m/s C. 0.0 m/s D. 0.03 m/s Since he started and ended at the same place his displacement was 0 so 10
Clicker question 5 Set frequency to BA Rank order the speeds at times 1, 2, and 3 from the slowest to the fastest A: V1 < V2 < V3 B: V2 < V1 < V3 C: V3 < V1 < V2 D: V3 < V2 < V1 E: V2 < V3 < V1 Speed = velocity, the magnitude of velocity. Velocity is the slope of the "x vs t" curve. At point 3, the slope is zero (it's flat). At point 2, it has a small (negative) slope. The speed there is small and positive. At point 1, it has a large positive slope. The speed there is large. So, V3 < V2 < V1 11
Displacement as a function of time Slight label change for average velocity gives where x 0 and t 0 are the starting position and time. Can set t 0 = 0. Solving for x gives If the velocity is constant: is the initial velocity where If the velocity is not constant, x = x 0 + where is the initial position at t=0 x 0 t 0 v(t)dt 12
Clicker question 6 Set frequency to BA A marathon runner runs at a steady 15 km/hr. When the runner is 7.5 km from the finish, a bird begins flying from the runner to the finish at 30 km/hr. When the bird reaches the finish line, it turns around and flies back to the runner, and then turns around again, repeating the back-and-forth trips until the runner reaches the finish line. How many kilometers does the bird travel? A: 10 km B: 20 km C: 15 km E: Not sure/impossible to decide The runner goes at 15 km/hr, and has to run 7.5 km to finish. How long does that take her? Well, 1/2 an hour (v*t = d, so t = d/v = 7.5 km / 15 km/hr =.5 hr) And during that half hour, the little bird is flying back and forth at a steady 30 km/hr. If it goes 30 km/hr * 1/2 an hour = 15 km, that's how far the bird went! 13
x vs t graphs A bicyclist goes 100 m in a straight line at constant speed in 10 s. How can we represent this motion? Can plot the position x versus x(m) 100 time t over the 10 seconds What is the average velocity and speed of the bicyclist? Note this is the slope of the above graph 0 10 t(s) Speed is the same 14
x(m) 100 Non-constant velocity Constant velocity v(m/s) x(m) 100 0 10 t(s) Non-constant velocity 10 0 10 t(s) v(m/s) 20 0 10 t(s) Slope of tangent line at each time 0 10 t(s) gives instantaneous velocity 15
Clicker question 7 Set frequency to BA Q. A train moves along a straight track. The graph shows the train position as a function of time. The graph shows that the train: A. speeds up the whole time B. slows down the whole time C. moves at constant velocity D. speeds up part of the time and slows down part of the time E. none of the above The slope is largest at t = 0 and continually decreases as time increases so the velocity is decreasing (but positive) 16
Acceleration Instantaneous velocity: For constant velocity case: so We define acceleration as change of velocity over time so in 1D Instantaneous acceleration is Acceleration is slope of velocity vs time graph For constant velocity, 17
x(m) 100 Position, velocity, acceleration vs t Constant velocity x(m) 100 v(m/s) a(m/s 2 ) 10 2 0 10 t(s) 0 10 t(s) 0 10 t(s) Constant acceleration 0 10 t(s) v(m/s) a(m/s 2 ) 20 2 0 10 t(s) 0 10 t(s) 18
Clicker question 8 Q. An object s velocity vs time graph is shown on the right. What best describes the car s acceleration vs time? a a A B Set frequency to BA t t a C a D t t E. None of these 19
Clicker question 9 Set frequency to BA A particle starts at the origin. Below is a graph of velocity vs. time. x is the AREA under the "v vs t" curve. The area under the triangle is (1/2) base*height = (1/2)*3 sec *6 m/s=9 m. Another approach is to observe that "straight line v vs t" means CONSTANT acceleration, so you What is the approximate position at t=3 can use delta x = (1/2)* a * t^2. sec? Here, a is the slope of the line, A: 3 m B: 6 m which is 6 m/s / (3 sec) = 2 m/s^2. C: 9 m D: 18 m So Delta x = (1/2)*(2 m/s^2)*(3 sec)^2 = 9 m. Same answer, Will E: None of these/not enough information. derive next lecture! 20
Summary Read Chapter 2 of H+R Motion along a Straight Line Register your Clickers Think about attending the talk on Saturday by Al Bartlett 21