Presentation on theme: "I recommend writing down items in these yellow boxes"— Presentation transcript:
1 I recommend writing down items in these yellow boxes Chapter 2: Motion in One DirectionMotion Is RelativeDisplacement vs. DistanceVelocity vs. SpeedVelocity: Average and InstantaneousAccelerationFree FallI recommend writing down items in these yellow boxes
2 “Motion Is Relative” – What does this mean? Motion of objects is always described as relative to something else. For example:Motion takes place over time and depends upon the frame of reference (precise location of objects in space).
3 Displacement – (your thoughts?) The length of the straight line drawn from its initial position to the object’s final position.Displacement: the change in position of an objectΔx = xf – xiΔx = Change in position on the x-axisΔ = Changexf = final position (on x-axis)xi (xo) = initial position (on x-axis)
4 Displacement xf xi Δx = xf – xi Δx = Change in position on the x-axisxixfΔx = xf – xi = 7.0 cm – 2.0 cm = 5.0 cm
5 Displacement xi xf Δx = xf – xi Δx = Change in position on the x-axisxixfΔx = xf – xi = 1.2 cm – 6.5 cm = -5.3cmPositive or negative depends on the direction
6 Positive or negative depends on the direction DisplacementΔy = yf – yiyfΔy = Change in position on the y-axisΔy = yf – yi= 10.0cm – 6.1 cm= 3.9cmyiPositive or negative depends on the direction
8 Displacement = Distance ??? xixfxixfxfDistance does not have positive or negative, it is the total distance traveled, NOT just the final outcome (xf – xi).
9 Velocity – (your thoughts?) Δx change in distanceΔt change in timevavg = = = m/sΔx xf – xiΔt tf – tivavg = =dtv = = m/sNOTE: For most problems, the initial values are going to be zero. We typically start at time zero, then begin. Our displacement also typically starts at a position of zero and moves a certain “distance”.
11 Velocity (v) is a description of Δx change in distanceΔt change in timev = = = m/sdtv = = m/sVelocity (v) is a description ofthe instantaneous speed of the objectwhat direction the object is moving (north, left, right, up, down, pos x-axis, neg x-axis)Vectors have magnitude and directionVelocity is a vector quantity. It hasmagnitude: instantaneous speeddirection: direction of object’s motion(typically positive and negative x-axis)
12 Speed (s) and Velocity (v) Constant speed is steady speed, neither speeding up nor slowing down.Constant velocity isconstant speed andconstant direction (straight-line path with no acceleration).Motion is relative to Earth, unless otherwise stated.
13 P.I.E.S.S ??? Δd change in distance Δt change in time v = = = m/s d t Example: A flying disc leave’s Mr. Pearson’s hand with a velocity of 15 m/s. Ignoring wind resistance, the disc travels for 10 seconds before hitting the ground. How far did the disc travel?Ignoring wind resistance =no friction = constant velocityv = 15 m/sdtv =d = t * vt = 10 s= 10 s * 15m/sd = ____ mv = 15 m/sd = 150 mP.I.E.S.S ???
14 Velocity t0 t1 t2 t3 Δx xf – xi Δt tf – ti vavg = = x0 = 0.0 cm 1) What is the displacement between x1 and x2?2) What is the average velocity over the entire movement from x0 to x3?3) What is the average velocity between t0 and t1?4) What is the average velocity between x2 and x3?
18 Instantaneous Velocity Instantaneous Velocity - is the velocity at any specific instant.Example:When you ride in your car, you may speed up and slow down - (this will affect your average velocity over a given distance and time)Your instantaneous speed is given by your speedometer (your speed at that moment).
19 Average SpeedThe entire distance covered divided by the total travel timeDoesn’t indicate various instantaneous speeds along the way.200 km2 h= 100 km/hExample: What is your average speed if you drive a total distance of 200 km in 2 hours?
20 Average Speed CHECK YOUR NEIGHBOR The average speed of driving 30 km in 1 hour is the same as the average speed of driving30 km in 1/2 hour.30 km in 2 hours.60 km in 1/2 hour.60 km in 2 hours.20
21 Average Speed CHECK YOUR ANSWER The average speed of driving 30 km in 1 hour is the same as the average speed of driving30 km in 1/2 hour.30 km in 2 hours.60 km in 1/2 hour.60 km in 2 hours.60 km2 hour=30 km1 hourExplanation:Average speed = total distance / timeSo, average speed = 30 km / 1 h = 30 km/h.SameNow, if we drive 60 km in 2 hours:Average speed = 60 km / 2 h = 30 km/h21
22 AccelerationFormulated by Galileo based on his experiments with inclined planes.
23 vo = velocity initial (if starting from rest vo = ZERO) Acceleration (a) - rate at which velocity changes over timeΔv vf - vo change in velocity m/sΔt tf - to time required for change sa = = = = = m/s2a = = = m/s2v m/st svf = velocity finalvo = velocity initial (if starting from rest vo = ZERO)tf = time finalto = time initialNOTE: Usually we don’t use tf - to just t = required timevo = velocity at the starting timeAt time = 0 sec
24 Your car’s acceleration is 2.5 km/h/s. Δv vf - vo change in velocity m/sΔt t time required for change sa = = = = = m/s2Example:You car’s speed right now is 40 km/h. (vo = 40 km/h)Your car’s speed 2 s later is 45 km/h. (vf = 45 km/h t = 2s )Your car’s change in speed is 45 – 40 = 5 km/h. (change = Δv)Δv vf - vo km/h – 40 km/h 5 km/hΔt t s sa = = = = = 2.5 km/h/sYour car’s acceleration is 2.5 km/h/s.
25 Δv vf - vo change in velocity m/s Δt t change in time s a = = = = = m/s2Example: A flying disc leave’s Mr. Pearson’s hand with a velocity of 15 m/s. After 3 seconds, wind resistance slows down the disc to 11 m/s. What is the acceleration of the disc? ,Change in velocity results in Accelerationvo = 15 m/svf - vota =11 m/s – 15 m/s3 s=vf = 11 m/s-4 m/s3 s=t = 3 sa = 1.3 m/s2The negative means that the acceleration is in the direction opposite to the motion of the object = resulting in slowinga = = = m/s2v m/st s
26 x vf = 8 m/s Δv vf - vo change in velocity m/s a = = = = = m/s2 Δt t change in time sa = = = = = m/s2Example: If a different disc has an acceleration of m/s2. What will the velocity be after 2 seconds of flight if Mr. Pearson releases the disc at 13 m/s.Don’t wait for the teacher, get to work vf = 8 m/sIs this answer REASONABLE? Explainxa = = = m/s2v m/st s
27 AccelerationWhen you are driving a car, what are three things you can do to cause acceleration?.Acceleration involves a change in velocity:A change in speed (faster or slower), orA change in direction, orBoth.NOTE: You do not feel velocityYou can feel a CHANGE in velocityYou DO feel ACCELERATION
28 Acceleration CHECK YOUR NEIGHBOR An automobile is accelerating when it isslowing down to a stop.rounding a curve at a steady speed.Both of the above.Neither of the above.28
29 Acceleration CHECK YOUR ANSWER An automobile is accelerating when it isslowing down to a stop.rounding a curve at a steady speed.Both of the above.Neither of the above.Explanation:Change in speed (increase or decrease) is acceleration, so slowing is acceleration.Change in direction is acceleration (even if speed stays the same), so rounding a curve is acceleration.Acceleration occurs due to a change in either speed or direction (or both).When a car slows down it changes its speed, so it is accelerating.When a car rounds a curve, although its speed is steady it is accelerating because it is changing direction.29
30 Acceleration CHECK YOUR NEIGHBOR Acceleration and velocity are actuallythe same.rates but for different quantities.the same when direction is not a factor.the same when an object is freely falling.30
31 Acceleration CHECK YOUR ANSWER Acceleration and velocity are actuallythe same.rates but for different quantities.the same when direction is not a factor.the same when an object is freely falling.Explanation:Velocity is the rate at which distance changes over time,Acceleration is the rate at which velocity changes over time.Δv vf - vo change in velocity m/sΔt t change in time sa = = = = = m/s2Δd change in distanceΔt change in timev = = = m/s31
32 Acceleration Galileo increased the inclination of inclined planes. Steeper inclines gave greater accelerations.When the incline was vertical, acceleration was max, same as that of the falling object.When air resistance was negligible, all objects fell with the same unchanging acceleration.Unchanging = CONSTANT AccelerationWhich ball hits the ground first?Which ball has the highest vf ?Which ball has the highest vo ?
33 Free FallFalling under the influence of gravity only - with no air resistanceFreely falling objects on Earth accelerate at the rate of 10 m/s/s, i.e., 10 m/s2a = 10 m/s2 = 9.8 m/s2 = m/s2 (sig figs)We say this is the acceleration due to gravity (g):a = g = 10 m/s = m/s2Text book vs Real World(easier math)
34 Free Fall—How Fast?a =ΔvtThe velocity acquired by an object starting from rest isSo, under free fall, when acceleration is a = g = 10 m/s2, the speed isv0 = 0 m/s before release: t0 = 0 sv1 = 10 m/s after 1 s t1 = 1 sv2 = 20 m/s after 2 s. t2 = 2 sv3 = 30 m/s after 3 s. t3 = 3 sAnd so on.Acceleration = Change of 10 m/s per second
35 Gravity WITH Friction Δv a = t In this situation g still equals 10 m/s2, but the motion of the object is no longer strait down. Here, we must include friction and trigonometry. In this example: a = 4 m/s2 NOT 10 m/s2v0 = 0 m/s before release: t0 = 0 sv1 = 4 m/s after 1 s t1 = 1 sv2 = 8 m/s after 2 s. t2 = 2 sv3 = 12 m/s after 3 s. t3 = 3 sAnd so on.Acceleration = Change of 4 m/s per second
36 Free Fall—How Fast?A free-falling object has a speed of 30 m/s at one instant. Exactly 1 s later its speed will bethe same.35 m/s.more than 35 m/s60 m/s.a =ΔvtΔ v = a*tvf = vo + Δvvf = vo + a*tvf = 30m/s + (10m/s2 * 1s)vo = 30 m/sThe object gains 10m/s after every second.So, at a given instance, if its speed is 30m/s, then a second later it will be 10m/s greater than that i.e. 30m/s + 10m/s = 40m/sSo, Its speed a second later is 40m/s which is more than 35m/s.vf = 30m/s + 10m/st = 1 svf = 40m/svf = ____ m/sa = g = 10 m/s2 Freely Falling36
37 Free Fall—How Far?The distance covered by an accelerating object -- starting from rest is:d = ½ *a*(t)2m = ½ *(m/s2)*(s)2Note that the seconds cancelSo, under free fall, when acceleration is 10 m/s2, the distance isat t = 1 s; d = ½ *a*(t)2 = ½ (10 m/s2) (1 s)2 = 5 mat t = 2 s; d = ½ *a*(t)2 = ½ (10 m/s2) (2 s)2 = 20 mat t = 3 s; d = ½ *a*(t)2 = ½ (10 m/s2) (3 s)2 = 45 mAnd so on
38 Free Fall—How Far? CHECK YOUR NEIGHBOR What is the distance covered of a freely falling object starting from rest after 4 s?4 m16 m40 m80 md = ½ *a*(t)2d = ½ *(10 m/s2)*(4 s)2d = (5 m/s2) * (16 s2)d = 80 mvo = 0 m/s (rest)t = 4 sd = ____ ma = g = 10 m/s2 Freely Falling38
39 2*d = (t)2 d = ½ *a*(t)2 a t = (2*d/a)1/2 t = (2*34 m / 10m/s2)1/2 EXAMPLE: A ball thrown straight up into the air begins to slow down (constant acceleration) until it stops in mid air at a height of 34 meters. How long does it take to reach this height?(NOTE: the same equation for free-fall is the same)d = 34 m2*d = (t)2ad = ½ *a*(t)2t = _____ sa = g = 10 m/s2t = (2*d/a)1/2Freely Fallingt = (2*34 m / 10m/s2)1/2t = (68 m / 10m/s2)1/2t = (6.8 s2)1/2t = 2.6 s d = 34 mIs this answer REASONABLE?
40 Free Fall—How Far? d = ½ *a*(t)2 Δd change in distance at t = 1 s; d = ½ *a*(t)2 = ½ (10 m/s2) (1 s)2 = 5 mat t = 2 s; d = ½ *a*(t)2 = ½ (10 m/s2) (2 s)2 = 20 mat t = 2.6 s; d = ½ *a*(t)2 = ½ (10 m/s2) (2.6 s)2 = 34 mat t = 3 s; d = ½ *a*(t)2 = ½ (10 m/s2) (3 s)2 = 45 mΔd change in distanceΔt change in timev = = = m/sdtv = = m/sΔv vf - vo change in velocity m/sΔt t change in time sa = = = = = m/s2d = ½ *a*(t)2 = ½ *(m/s2)*(s)2 = ma = g = 10 m/s2 = 9.8 m/s2 (for freely falling objects)
41 d = ½ *a*(t)2 vf = vo + a*t a = g = 10 m/s2 Δv a = t Figure 3.8t3 = 3 s v3 = 0 m/st2 = 2 s v2 = 10 m/st4 = 1 st4 = 4 s v4 = -10 m/s(negative = direction)t1 = 1 s v1 = 20 m/st5 = 2 st5 = 5 s v5 = -20 m/st0 = 0 s v0 = 30 m/st6 = 3 st6 = 6 s v6 = -30 m/sa = g = 10 m/s2d = ½ *a*(t)2a =Δvtvf = vo + a*tt7 = 7 s v7 = -40 m/st7 = 4 s