10Test your understanding In which of these situations would the average velocity vector vav over an interval be equal to the instantaneous velocity v at the end of interval:A body moving along a curved path at constant speedA body moving along a curved path and speeding upA body moving along a straight line at constant speedA body moving along a straight line and speeding up.
11example v = dr/dt = 2bt i + 3ct2 j If r = bt2i + ct3j t = Where b and c are positive constants, when does the velocity vector make an angle of 45.0o with the x- and y-axes?v = dr/dt = 2bt i + 3ct2 j= 12b3ct =
16Average acceleration vs. instantaneous acceleration
17Example 3.2Find the components of the average acceleration in the interval from t = 0.0 s to t = 2.0 s.Find the instantaneous acceleration at t = 2.0 s, its magnitude and its direction.
18Parallel and perpendicular components of acceleration
19The effect of acceleration’s directions When acceleration is perpendicular to particle’s velocity: velocity’s magnitude does not change, only its direction changes, particle moves in a curved path at constant speed.When acceleration is parallel to particle’s velocity: velocity’s magnitude changes only, its direction remains the same, particle moves in a straight line with changing speed.
21Example 3.3Given:Find the parallel and perpendicular components of the instantaneous acceleration at t = 2.0 s
22Test your understanding 3.2 A sled travels over the crest of a snow-covered hill. The sled slows down as it climbs up one side of the hill and gains speed as it descends on the other side. Which of the vectors (1 though 9) in the figure correctly shows the direction of the sled’s acceleration at the crest? Choice 9 is that the acceleration is zero.)
253.3 projectile motionA projectile is any body that is given an initial velocity and then follows a path determined entirely by the effects of gravitational acceleration and air resistance.Example: a batted baseball, a thrown football, a package dropped from an airplane, a bullet shot from a rifle.The path of a projectile is called a trajectory.
41Example 3.5Let’s consider again the skier in example 3.4. What is her acceleration at points G, H, and I after she flies off the ramp? Neglect air resistance.The acceleration at points G, H, I are the same:ax = 0; ay = -9.8 m/s2
42Example 3.6:a body projected horizontally A motorcycle stunt rider rides off the edge of a cliff. Just at the edge his velocity is horizontal, with magnitude 9.0 m/s. find the motorcycle’s position, distance and velocity from the edge of the cliff, and velocity after 0.50 s.
43x = vocosα0 ∙t = (9.0 m/s)cos0o(0.5 s) = 4.5 m Position:α0 = 0o;vo = 9.0 m/s; voy = 0; t = 0.5 sx = vocosα0 ∙t = (9.0 m/s)cos0o(0.5 s) = 4.5 my = vosinα0 ∙t - ½ gt2= - ½ (9.8 m/s2)(0.5 s)2 = -1.2 mdistance:r = √ (-1.2 m)2r = 4.7 m
49You toss a ball from your window 8. 0 m above the ground You toss a ball from your window 8.0 m above the ground. When the ball leaves your hand, it is moving at 10.0 m/s at an angle of 20o below the horizontal. How far horizontally from your window will the ball hit the ground? Ignore air resistance.
50Given: y = - 8.0 m; α0 = - 20o; v0 = 10.0 m/s; Unknown: x = ?To find t, we use-8.0 m = (10.0 m/s)sin(-20o)t – ½ (9.8 m/s2)t2t1 = -1.7 s; t2 = 0.98 sx = (10.0 m/s)cos(-20o)(0.98 s) = 9.2 m
51A monkey escapes from the zoo and climbs a tree A monkey escapes from the zoo and climbs a tree. After failing to entice the monkey down, the zookeeper fires a tranquilizer dart directly at the monkey. The clever monkey lets go at the same instant the dart leaves the gun barrel, intending to land on the ground and escape. Show that the dart always hits the monkey, regardless of the dart’s muzzle velocity (provided that it gets to the monkey before he hits the ground).
55If dart hits the monkey, the position of the dart and the position of the monkey must be the same at time t.For the dart:For the monkey:xdart = d = (v0cosα0)tydart = (v0sinα0)t – ½ gt2xmonkey = dymonkey = h – ½ gt2For the dart:xdart = d = (v0cosα0)t t = d / (v0cosα0)ydart = (v0sinα0)(d /(v0cosα0)) – ½ gt2ydart = (dtanα0) – ½ gt2Since tanα0 = h / dydart = h – ½ gt2 = ymonkey
56Check your understanding 3.3 In example 3.10, suppose the tranquilizer dart has a relatively low muzzle velocity so that the dart reaches a maximum height at a point before striking the monkey. When the dart is at point P, will the monkey beAt point A (higher than P)At point B (at the same height as P)At point C (lower than P)?
57v Sail boat moving at v relative to the ground Sail boat moving at v’ relative to the ship. v’ = ?ship moving at V relative to the ground
58vAB = vAground – vBground Sail boat moving at v relative to the groundSail boat moving at v’ relative to the ship. v’ = ?ship moving at V relative to the groundRelative velocity of object A to object BAt any time tr = r’ + Rv∙t = v’∙ t + V∙tv’ = v - VvAB = vAground – vBgroundNote:vAB = vAground + vgroundB
66Do now (ap exam 2009) All the following are units of power except WattsJoules per secondElectron volts per secondNewton meters per secondKilogram meters per second72% correct
67do nowA 2 kg ball collides with the floor at an angle θ and rebounds at the same angle and speed as shown above. Which of the following vectors represents the impulse exerted on the ball by the floor?A.θθB.-v1C.D.v2v1E.Impulse = change in momentum = m∆vImpulse has the same direction as v2 – v1 or v2 + (-v1)
68Do now (ap 2009 exam)A projectile is launched from level ground with an initial speed v0 at an angle θ with the horizontal. If air resistance is negligible, how long will the projectile remain in air? (represent t in terms of v0, θ, and g)t = 2v0sin θ / g 50% correct
69Do now (ap exam 2009)If air resistance is negligible, what is the speed of a 2 kg sphere that falls from rest through a vertical displacement of 0.2 m?2 m/s; 70% correct