Motion: speed, velocity and acceleration
Our first example of a “time rate of change,” how quickly slowly something changes in time.
Most of the stuff you see around you is not in motion. Our reference for motion is the surface of the earth and those things at rest relative to it—buildings, bridges, telephone poles, streets, cities, etc.
You’re already familiar with the concept of speed in miles per hour, from reading a speedometer and from seeing speed limit signs. A speedometer registers what’s called instantaneous speed, the speed at one “instant” of time. On your way to work or school, you travel at many different instantaneous speeds, including zero miles per hour when you’re stopped.
Your average speed is the distance you drive to work or school divided by the time it takes to get there. So to find the average speed, you just divide distance traveled by the time of travel,
v = d / t.
Stopping at a stop sign or traffic light is going to increase time, making average speed lower.
EXAMPLE 1: distance = 7.5 miles, time 15 min. Did not have to stop at any stop lights. Want mph, how to get it? 15 min (1 hr/60 min) = 0.25 hr. ` v = 7. 5 mi/ .25 hr. = 30 mph.
EXAMPLE 2: same distance but this time you get stopped at three red lights for 20 sec each. 3 X 20 = 60 sec =1 min. Time to travel 7.5 mi is now . . . 16 min X (1hr/60min) = .27 hr, so we have 7.5 mi / .27 hr = 28 mph.
If you have 15 min to get to work or school, you’ll be upping your average speed by going over 30 after the first stoplight. Then you might actually catch the next one green. Or may have to wait again for less time.
Def. of motion “When an object is undergoing a continuous change in its position we say the object is moving or is in motion.” Change in position relative to what? All those things that are stationary, the totality of which we call a reference frame or rest frame.
“A journey of a thousand miles begins with one step.” The step must be in some direction! When direction must be specified, the quantity becomes a vector. Displacement is the name of the vector representing a distance in some given direction.
Motion can be either constant speed, constant velocity, or accelerated motion. Acceleration itself can be constant or increasing or decreasing but constant acceleration is the only case dealt with in beginning physics classes.
Until Galileo did his experiments with falling and rolling objects, acceleration was not understood and people just thought of “motion.” Galileo did QUANTITATIVE OBSERVATIONS (measurements) that related distance to time (made and used a water clock). Galileo “discovered” acceleration.
Can now calculate average speed, but how do you calculate velocity and acceleration?
First thing to know is they are vectors. In physics, a vector is something that has magnitude and direction. A vector quantity is represented by an arrow. The length of the arrow is the magnitude. The direction is the direction the arrow is pointing.
VELOCITY.
The length of a velocity vector has a special name. SPEED. If you just want to know how fast something is going and don’t care or already know what direction it’s going in, you talk about its speed. Our study of motion starts with speed.
RECALL what came up in first class—the measurements made that day. Speed was one of them. Since you’re most familiar with speed in the context of looking at your speedometer and making sure you’re not going too fast, let’s talk about what a speedometer measures. That’s called instantaneous speed.
Instantaneous speed = speed at one instant of time.
Instantaneous VELOCITY = speed and direction at one instant of time.
If you’re talking on you cell phone (probly shouldn’t be) and you tell someone you’re driving south on Hazel Street at 40 miles an hour, you’re telling them your instantaneous velocity. (Note that this doesn't tell the person where you are on Hazel St.)
Your instantaneous speed or velocity is not by itself going to tell you if you’re going to get to work or class on time. Not unless you start off at one particular speed and don’t stop or speed up all the way there. Your average speed is what determines how fast you get somewhere.
AVERAGE SPEED = distance traveled/ time to travel that distance. ` v = D d/D t. Usually you can just write this as ` v = d/t.
Also have AVERAGE VELOCITY. First need to have a concept called displacement. Again we have a vector quantity. This time the magnitude of the vector is distance from the starting point, and direction is along a straight line from starting point to ending point.
AVG VELOCITY = displacement/ total travel time. See example in book. Fig. 2.4 Beyond the scope of this class. But should know: if a car returns to its starting point, then the displacement is zero, so the avg. velocity is zero too.
ACCELERATION: ` a = D v/D t = (vf – vi)/ (tf – ti) change in velocity divided by change in time. Usually can write it as just ` a = D v /t, and if vi = 0, can write a =vf/t. vf =at.
EXAMPLES ARE GRAVITY, CIRCULAR MOTION, AND PROJECTILE MOTION (object falling due to gravity).
Near the surface of Earth, objects fall with constant acceleration, 9.8 m/s^2.
How fast will object be traveling when it hits the ground dropped from a certain height? d= ½ gt^2 has time in it. Vf = gt gives v speed in terms of time. We want speed in terms of distance fallen. What to do? Solve one equation for t and substitute into the other equation “eliminate the unknown between the two equations.”
T=vf/g, d= ½ g (vf/g)^2 = ½ vf^2/g. 2dg=vf^2,
EXAMPLE ONE: How high to hit ground at 60 mph? D = vf^2 / 2g. 60mph (1ft/s / .682 mph) = 88 ft/s. d= 88ft/s ^2 / 2(32 ft/s^2) = 121 ft. Simmons Bank. Wear a seat belt!
EXAMPLE TWO: What is speed if jump from 12 feet? Vf= (2X12ftX32ft/s^2)^ ½ = sqrt(24X32) ft/s = sqrt288 = 17 ft/s (.682mph/1ft/s) = 11.6 mph.
Centripetal acceleration: ac = v^2 / r Example: exercise 17.
Projectile motion: constant velocity in x direction, constant acceleration “g” in y direction. See Fig 2.15.
Figure 2.17 shows 45 degrees as max radius, and shows “complementary angles” of 30 and 60 give same range. “Complementary” means they add to 90 degrees. This word is different in spelling and meaning from the word "complimentary." Keep that in mind, or you may wind up complementing someone when you meant to compliment him or her. Or vice-versa.
