How do We Measure Speed (2.1) Velocity Versus Speed Definition 1 Scalars are quantities that are fully described by a magnitude (or numerical value) alone. Definition 2 Vectors are quantities that are fully described by both a magnitude and a direction. Definition 3 Speed is a scalar quantity that refers to how fast an object is moving. Definition 4 Velocity is a vector quantity that refers to the rate at which an object changes its position. Example 1 An example of a speed is 55mph. An example of a velocity is 55mph North. Average Velocity Definition 5 If s(t) is the position of an object at time t, then the average velocity of the object over the interval a ≤ t ≤ b is given by Average Velocity = ∆s ∆t = s(b)−s(a) b−a In words, the average velocity of an object over an interval is the net change in position during the interval divided by the change in time. Example 2 The trajectory of a grapefruit 1 Table: Height of the t (sec) 0 1 2 3 4 5 6 Grapefruit vs Time y (feet) 6 90 142 162 150 106 30 1. Compute the average velocity of the grapefruit over the interval 1 ≤ t ≤ 3. What is the significance of the sign of your answer? 2. Compute the average velocity of the grapefruit over the interval 4 ≤ t ≤ 5. What is the significance of the sign of your answer? Note 1 In calculus we use the following notation: Average Velocity = s(a+h)−s(a) h Example 3 Graph: Height of the Grapefruit vs Time 2 Note 2 The average velocity over any time interval a ≤ t ≤ b is the slope of the line joining the points on the graph of s(t) corresponding to t = a and t = b. Example 4 Calculate the Average Velocity, over the time interval from 1 second to 2 seconds, of the grapefruit whose trajectory is given by the graph above. Solution. average velocity = s(2)−s(1) 2−1 average velocity = = = 52 fst 142−90 1 s(1+1)−s(1) 1 = 142−90 1 = 52 fst Problem 1 The distance, s, a car has traveled on a trip is shown in the table as a function of the time, t, since the trip started. Find the average velocity between t = 2 and t = 5. t (hrs) 0 1 2 3 4 5 s (kms) 0 45 135 220 300 400 Problem 2 The table gives the position of a particle moving along the xaxis as a function of time in seconds, where x is in angstroms. What is the average velocity of the particle from t = 2 to t = 8? t 0 2 4 6 8 x(t) 0 72 92 144 180 3 Instantaneous Velocity Definition 6 The instantaneous velocity is the slope of the distance function at a point. Average Velocity= s(a+h)−s(a) h Instantaneous Velocity (at t = a) = limh→0 s(a+h)−s(a) h Problem 3 Find the average velocity over the interval 0 ≤ t ≤ .2, and estimate the velocity at t = 0.2 of a car whose position, s, is given by the following table: t (sec) 0 .2 .4 .6 .8 1.0 s (feet) 0 .5 1.8 3.8 6.5 9.6 Using Limits to Compute the Instantaneous Velocity Example 5 Calculate the instantaneous velocity for s(t) = t2 at t = 3. Solution. 4 METHOD 1. Estimate the limit numerically: h -.1 -.01 -.001 .001 .01 .1 (3+h)2 −9 h 5.9 5.99 5.999 6.001 6.01 6.1 METHOD 2. Use algebra to find the limit: s(a + h) − s(a) h→0 h lim (a + h)2 − a2 h→0 h 2 a + 2ah + h2 − a2 = lim h→0 h 2ah + h2 = lim h→0 h h (2a + h) = lim h→0 h = lim 2a + h = 2a = lim h→0 Now we use the fact that a = 3 to find that the instantaneous velocity is 6. H Problem 4 In a time of t seconds, a particle moves a distance of s meters from its starting point, where s = 3t2 . (a) Find the average velocity between t = 1 and t = 1 + h if: (i) h = 0.1 5 (ii) h = 0.01 (iii) h = 0.001 (b) Use your answers to part (a) to estimate the instantaneous velocity of the particle at time t = 1. Problem 5 Use algebra to evaluate limh→0 Problem 6 Estimate limh→0 ues of h. cos(h)−1 h (1+h)3 −1 . h by substituting smaller and smaller val- Problem 7 The graph of f (t) in the figure gives the position of a particle at time t. Estimate the following quantities: A. average velocity between t = 1 and t = 3 B. average velocity between t = 5 and t = 6 C. instantaneous velocity at t = 1 D. instantaneous velocity at t = 3 6 E. instantaneous velocity at t = 5 F. instantaneous velocity at t = 6 Problem 8 A ball is tossed into the air from a bridge, and its height, y (in feet), above the ground t seconds after it is thrown is given by the equation y = f (t) = −16t2 + 50t + 36. (a) How high above the ground is the bridge? (b) What is the average velocity of the ball for the first second? (c) Approximate the velocity of the ball at t = 1 second. (d) Graph f , and determine the maximum height the ball reaches. What is the velocity at the time the ball is at the peak? (e) Use the graph to decide at what time, t, the ball reaches its maximum height. 7