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1A motorboat going downstream overcame a raft at a point A; τ = 60 min later it turned back and after some time passed the raft at a distance l = 6.0 km from the point A. Find the flow velocity assuming the duty of the engine to be constant.
2A point traversed half the distance with a velocity v0. The remaining part of the distance was covered with velocity v1 for half the time, and with velocity v2 for the other half of the time. Find the mean velocity of the point averaged over the whole time of motion.
3A car starts moving rectilinearly, first with acceleration w = 5.0 m/s2 (the initial velocity is equal to zero), then uniformly, and finally, decelerating at the same rate w, comes to a stop. The total time of motion equals τ = 25 s. The average velocity during that time is equal to <v> = 72 km per hour. How long does the car move uniformly?
4A point moves rectilinearly in one direction. Fig. 1.1 shows the distance s traversed by the point as a function of the time t. Using the plot find: (a) the average velocity of the point during the time of motion; (b) the maximum velocity; (c) the time moment t0 at which the instantaneous velocity is equal to the mean velocity averaged over the first t0 seconds.
5Two particles, 1 and 2, move with constant velocities v1 and v2. At the initial moment their radius vectors are equal to r1 and r2. How must these four vectors be interrelated for the particles to collide?
6A ship moves along the equator to the east with velocity v0 = 30 km/hour. The southeastern wind blows at an angle φ = 60° to the equator with velocity v = 15 km/hour. Find the wind velocity v* relative to the ship and the angle φ* between the equator and the wind direction in the reference frame fixed to the ship.
7Two swimmers leave point A on one bank of the river to reach point B lying right across on the other bank. One of them crosses the river along the straight line AB while the other swims at right angles to the stream and then walks the distance that he has been carried away by the stream to get to point B. What was the velocity u of his walking if both swimmers reached the destination simultaneously? The stream velocity v0 = 2.0 km/hour and the velocity v* of each swimmer with respect to water equals 2.5 km per hour.
8Two boats, A and B, move away from a buoy anchored at the middle of a river along the mutually perpendicular straight lines: the boat A along the river, and the boat B across the river. Having moved off an equal distance from the buoy the boats returned. Find the ratio of times of motion of boats τA/τB if the velocity of each boat with respect to water is η = 1.2 times greater than the stream velocity.
9A boat moves relative to water with a velocity which is n = 2.0 times less than the river flow velocity. At what angle to the stream direction must the boat move to minimize drifting?
10Two bodies were thrown simultaneously from the same point: one, straight up, and the other, at an angle of θ = 60° to the horizontal. The initial velocity of each body is equal to v0 = 25 m/s. Neglecting the air drag, find the distance between the bodies t = 1.70 s later.
11Two particles move in a uniform gravitational field with an acceleration g. At the initial moment the particles were located at one point and moved with velocities v1 = 3.0 m/s and v2 = 4.0 m/s horizontally in opposite directions. Find the distance between the particles at the moment when their velocity vectors become mutually perpendicular.
12Three points are located at the vertices of an equilateral triangle whose side equals a. They all start moving simultaneously with velocity v constant in modulus, with the first point heading continually for the second, the second for the third, and the third for the first. How soon will the points converge?
13Point A moves uniformly with velocity v so that the vector v is continually "aimed" at point B which in its turn moves rectilinearly and uniformly with velocity u< v. At the initial moment of time v _|_ u and the points are separated by a distance 1. How soon will the points converge?
14A train of length l = 350 m starts moving rectilinearly with constant acceleration w = 3.0*10-2 m/s2; t = 30 s after the start the locomotive headlight is switched on (event 1), and τ = 60 s after that event the tail signal light is switched on (event 2). Find the distance between these events in the reference frames fixed to the train and to the Earth. How and at what constant velocity V relative to the Earth must a certain reference frame K move for the two events to occur in it at the same point?
15An elevator car whose floor-to-ceiling distance is equal to 2.7 m starts ascending with constant acceleration 1.2 m/s2; 2.0 s after the start a bolt begins falling from the ceiling of the car. Find: (a) the bolt s free fall time; (b) the displacement and the distance covered by the bolt during the free fall in the reference frame fixed to the elevator shaft.
16Two particles, 1 and 2, move with constant velocities v1 and v2 along two mutually perpendicular straight lines toward the intersection point O. At the moment t = 0 the particles were located at the distances l1 and l2 from the point O. How soon will the distance between the particles become the smallest? What is it equal to?
17From point A located on a highway (Fig. 1.2) one has to get by car as soon as possible to point B located in the field at a distance l from the highway. It is known that the car moves in the field η times slower than on the highway. At what distance from point Done must turn off the highway?
18A point travels along the x axis with a velocity whose projection vx is presented as a function of time by the plot in Fig. 1.3. Assuming the coordinate of the point x = 0 at the moment t = 0, draw the approximate time dependence plots for the acceleration wx, the x coordinate, and the distance covered s.
19A point traversed half a circle of radius R = 160 cm during time interval τ = 10.0 s. Calculate the following quantities averaged over that time: (a) the mean velocity <v>; (b) the modulus of the mean velocity vector |<v>|; (c) the modulus of the mean vector of the total acceleration |<w>| if the point moved with constant tangent acceleration.
20A radius vector of a particle varies with time t as r = at (1 - αt), where a is a constant vector and α is a positive factor. Find: (a) the velocity v and the acceleration w of the particle as functions of time; (b) the time interval Δt taken by the particle to return to the initial points, and the distance s covered during that time.
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