2. Solve Example 2 of the preceding section if the balloon is descending. 3. A high jumper raises his center of gravity 3 feet. How long is he off the ground, and with what velocity does he light? 4. A baseball dropped from one of the windows of the Washington monument, 500 feet from the ground, has been caught. Compare the velocity with which it struck the catcher's hand with the velocity of 120 feet per second which is said to be the maximum velocity that a pitcher has imparted to a ball. 5. A man descending in an elevator whose velocity is 10 feet per second drops a ball from a height of 6 feet above the floor. How far will the elevator descend before the ball strikes the floor of the elevator? 6. In the preceding problem, suppose the elevator is ascending instead of descending. 7. A balloon is ascending with a velocity of 24 feet per second when a ball is dropped from it. The ball reaches the ground in 5 seconds. Find the height of the balloon when the ball was dropped. Determine the highest point that the ball reached. 8. An automobile reduces its speed from 35 miles an hour to 20 miles an hour in 8 seconds. If the retardation is uniform how much longer will it be before it will come to rest, and how far will it travel in this length of time? 9. How high will a ball rise if thrown vertically upward with an initial velocity of 60 feet per second? 10. A street car in going from one stop to another 400 feet distant is uniformly accelerated at the rate of 2 feet per second per second for a distance of 320 feet and then brought to rest with a uniform retardation. Find the time it took to go the 400 feet. 108. Motion in a Plane. If a particle moves along a curve in a plane, two rectangular axes are chosen and the position of the particle at any time determined by means of the coördinates of the particle. If the coördinates of the particle on the axes are given by the equations x = f(t) and y = F(t) the position of the particle in the plane is determined. (1) The discussion of the motion of a particle in a plane is thus resolved into the discussion of two motions along straight lines. The components of the velocity, obtained by differentiating equations (1) are They are represented in Fig. 178 by directed lines parallel to the axes. Since the components are perpendicular the magnitude of the resultant velocity, v, is given by the equation and the direction of the velocity can be found from the equation tan 0 == Vy/Vx. (4) The components of the acceleration, obtained by differentiating equations (2) are They are represented in Fig. 179 by directed lines parallel to the axes. The magnitude of the resultant acceleration, a, is given by the equation α = √a2x + a2y and its direction can be found by means of the equation (6) (8) EXAMPLE 1. If a particle moves in accordance with the law x 13, y = t2, find the equation of the path, the position of the particle when t = 2, and the magnitude and direction of v and of a at this point. From the first equation t = x and hence y = (x3)2 = x}, which is the equation of the path shown in Fig. 180. Differentiating the given equa tions with respect to t we obtain = Substituting t 2 in equations (2), (3), (5), (6), we find v = 12.64, 0 = 18°.43, a = 12.17, 9°.46, at the point where x = = 8 and y = 4. The inverse problem of determining the path of a particle, given the equations of the component accelerations, is illustrated in EXAMPLE 2. Find the equation of the path of a projectile fired at an angle of 30° to the horizontal with a muzzle velocity of 1200 feet per second. components of the initial velocity are 1200 cos 30°, and 1200 sin 30°, respectively. If the resistance of the air be disregarded there will be no horizontal acceleration. From the instant the projectile leaves the gun gravity is acting on it vertically in the direction opposite to the positive direction on the y-axis. Hence the acceleration equations of the motion are Integrating (3), x = 1200 cos 30°t + C3 and y = - 16t2+1200 sin 30°t + C1. To determine the constants we note that when t = 0, x = 0, y = 0, and therefore C3 = 0, and C1 = 0. These equations give the position of the particle at any time. Solving the first of these equations for t and substituting in the second we find the equation of the path to be 16 y x2+tan 30°•x, 12002 cos2 30° (5) which is the equation of a parabola. To find the range, let y = 0 in (5) and solve for x. At the highest point v = 0, hence from (3), 0 = = 7.38 miles. 1200 sin 30° = Therefore the highest point is reached when t 18 seconds. Substituting this value of t in (4), we obtain for the maximum height attained 1. In the following exercises find the equation of the path in terms of x and y by eliminating t from the given equations, plot the path, find the magnitudes and directions of v and a for the given value of t, and at the point on the path corresponding to this value draw lines representing v and a in magnitude and direction. 2. Find the equation of the path of a projectile fired at an angle of 20° to the horizontal with a muzzle velocity of 1500 feet per second; find the range of the projectile, the maximum height attained and plot the path. 3. Find the equation of the path of a projectile fired at an angle of 0° to the horizontal with an initial velocity of vo feet per second. Find the range and the maximum height attained. Find vo if the range is 24 miles for 0 = 45°. 4. A bomb is dropped from an aeroplane 8000 feet high moving horizontally at a velocity of 120 miles an hour. Determine how far the bomb will fall from the point on the ground directly below the aeroplane at the instant it was dropped. Suggestion: If the point on the ground is chosen as the origin with the axes horizontal and vertical, the initial conditions are as given in the table 5. A bomb is dropped from an aeroplane h feet high and moving with a velocity of vo feet per second. If d represents the distance from a point on the ground directly below the aeroplane when the bomb was dropped to the point where it strikes the ground, find d as a function of h and v。. Plot the graph of d as a function of vo, h being constant, also of d as a function of h, vo being constant. In the first case what is the effect on d of doubling vo, and in the second case of doubling h? 6. A ball rolls off a roof inclined at 30° to the horizontal. If it starts 20 feet from the eaves, which are 30 feet from the ground, where will the ball strike the ground? 7. The path of a particle is given by the equations x = t2 and y = 2t. Plot the path, find the acceleration when t 1, and the components of the acceleration along the tangent and normal lines to the path at the point in question. 8. Find the range of a rifle ball fired horizontally from a point 5 feet above the ground if the initial velocity is 2000 feet per second. How much is the deviation from the horizontal at 100 yards? 200 yards? 9. A particle moves in a circle whose center is at the origin of a rectangular system of axes. The angle (in radians) through which it turns from an initial position at rest on the x-axis is given by the equation 0 = t2t. Find the angular velocity w D0 and the angular acceleration α = Diw when t = 3. = 10. A wheel revolving at the rate of 120 revolutions per second is retarded uniformly so that in 3 seconds, w = 90 revolutions per second. How long before the wheel will come to rest, and where will a point P on the rim be if when t O the angle for P is 0 = = 0? 109. Volume of a Right Prism. A prism is a solid bounded by two congruent polygons lying in parallel planes with their corresponding sides parallel and by the parallelograms determined by the pairs of corresponding sides of the polygons. The other sides of these parallelograms are called the lateral edges of the prism, the polygons are called the bases, and the parallelograms the lateral faces of the prism. A right prism is one whose lateral edges are perpendicular to the planes of the bases. |