Vectors Kinematics - 1 dimension Kinematics - 3 dimensions magnitudeA = A = A X2 - X1 . Ax Ar Vave= V ave A + B =B+A t2 -t At At (A + B) + C = A+(B+C) Ax dx Ar dr v = lim - v = lim At->0 At dt At-0 At dt A = Ai + A j + A,k V2 - VI. Av Ax dx a ave v. = lim A = A +A, t2-t1 At At-0 At dt A . B = AB COSO AB Av dv Ay v. = lim ay a = lim At-70 At At-0 At dt A . B = AB + AB, + A, A, dt v = Vo + at Av |A x B| = ABsinGAB dave At AXB = -B XA x = Xo + Vot+- at2 AV dv 1 x j = k, j xk =1, 2 a = lim V2 = Vo + 2a(x-X0) At-0 At dt k xi = j dv, Vo tv a = lim x-XO =- At-0 dt 2 Newton's Laws Av, dv 15. EF = 0 a = lim Constants and Math At-0 At dt 2nd. EF = ma g = 9.8(m/s' ) a centr = a= V R 3rd. F12 = -F21 ax- +bx + c= 0 2TER SF = max V = - b+Vb2 - 4ac T EF, = may - x= 2a Forces EF1 = ma =m- sin2 0 + cos2 0 =1 fx = UN R sin = opp / hyp f, SUN F. = mg cos = adj / hyp D = = CPAv2 sin 0 Fcentr = mal = mv2/r tan 0 = cos O Fa = mg W = mg1.When a high-speed passenger train traveling at vi: = 142 km/h rounds a bend, the engineer is shocked to see that a locomotive has improperly entered onto the track from a siding l' \" ' and is a distance D = 786 m ahead 1| I i . - m (see the gure). The locomotive is Lulu\" "' -" I111 mnnliu' moving at a constant VL = 29 km/h. The engineer of the passenger train immediately applies the brakes. Assume that a +x axis extends in the direction of motion and that the acceleration is constant. A. (10 pts) If a collision is to be just avoided, then the passenger train and the locomotive would barely touch each other. What is the velocity along the x axis of the passenger train at that point (in m/s)? Justify your answer. B. (10 pts) If the collision is just avoided, how long after applying the brakes does the passenger train barely touch the locomotive? C. (10 pts) What must be the constant acceleration along the X axis if a collision is to be just avoided? (magnitude in inn/52 and direction) 2. A cat sits on a merry-go-round platform turning with uniform y circular motion. At time t1 = 1.60 s, the cat's velocity :21 = (2.90 g) E + (2.60 g, measured on a horizontal x-y coordinate system. At t: = 5.90 5, its velocity is 192 = (2.90 3 i + (2.60 3;\". A. (10 pts) What is the value of the radius, r, where the cat is located? 3 t1 B. (10 pts) What is the magnitude and direction of the centripetal acceleration of the cat at time Q? {for the direction either describe the direction of the acceleration OR specify it in terms if unit vectors 3 and j') C. (10 pts) Ifthe only force holding the cat in place is the static friction between the cat and the merrygo-round platform, then what is the minimum #5 that will keep the cat in place without slipping on the platform? 3. In the gure, a block of mass m = 5.00 kg is sent sliding up a plane inclined at 13 = 37.0\" with initial speed v0 = 4.40 m/s while a horizontal force F of magnitude 50.0 N acts on it. The coefcient of kinetic friction between block and plane is 0.330. Use 9 = 9.8 m/sz as the acceleration due to gravity. A. (10 pts) Draw the free body diagram for the block while it is sliding up the plane, 9 and briey describe all the forces acting on it, and their directions. B. (10 pts) What is the net acceleration of the block as it is sliding up the plane? {Magnitude and direction, i.e. up or down the slope) C. (10 pts) If the block initially started at the bottom of the slope, what is the maximum height {vertically} that it reaches above the bottom of the slope? D. (10 pts} Assume ,us > ,uk. When the block reaches its highest point will it remain at rest or slide back down the slope? Justify your answer. If you said it remains at rest then what is the direction of the static frictional force (i.e. up or down the slope)? \"111