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Prove that the trajectory of a projectile is parabolic, having the form
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- A hiker walks from (x1, y1) = (4.00 km. 3.00 km) to (x2, y2) = (3.00 km, 6.00 km), (a) What distance has the traveled? (b) The hiker desires to return to his starting point. In what direction should he go? (Give the angle with respect to due cast.) (See Sections 3.2 and 3.3.)arrow_forwardA fish swimming in a horizontal plane has velocity i = (4.00 î + 1.00 ĵ) m/s at a point in the ocean where the position relative to a certain rock is i = (12.0 î − 2.60 ĵ) m. After the fish swims with constant acceleration for 15.0 s, its velocity is = (25.0 î − 1.00 ĵ) m/s. (a) What are the components of the acceleration of the fish? ax = ay = (b) What is the direction of its acceleration with respect to unit vector î? Draw coordinate axes on a separate piece of paper, and then add the acceleration vector with its tail at the origin. Write the numerical values for the x and y components and then use this drawing to determine the angle.° counterclockwise from the +x-axis(c) If the fish maintains constant acceleration, where is it at t = 28.0 s? x = m y = m In what direction is it moving? ° counterclockwise from the +x-axisarrow_forwardIn the figure, a ball is launched with a velocity of magnitude 6.00 m/s, at an angle of 44.0° to the horizontal. The launch point is at the base of a ramp of horizontal length d1 = 6.00 m and height d2 = 3.60 m. A plateau is located at the top of the ramp. (a) Does the ball land on the ramp or the plateau? When it lands, what are the (b) magnitude and (c) angle of its displacement from the launch point?arrow_forward
- The position r→ of a particle moving in an xy plane is given by r→=(3.00t^3−7.00t)i^+(6.00−2.00t^4)j^ with r→ in meters and t in seconds. In unit-vector notation, calculate(a)r→, (b)v→, and (c)a→ for t = 2.00 s. (d) What is the angle between the positive direction of the x axis and a line tangent to the particle's path at t = 2.00 s? Give your answer in the range of (-180o; 180o).arrow_forwardA football is placed at rest on the field with initial coordinates x0 = 0 and y0 = 0. The ball is then kicked and thereby given an initial velocity of magnitude v0 = 23.6 m/s and direction θ0 = 36.3o above the horizontal. Ignore the effects of air resistance. Initial horizontal component of the velocity v0x = Initial vertical component of the velocity v0y = Horizontal component of the acceleration ax = Vertical component of the acceleration ay = Enter the equation for the horizontal position versus time, x(t) = Enter the equation for the vertical position versus time, y(t) = Enter the equation for the horizontal velocity versus time, vx(t) = Enter the equation for the vertical velocity versus time, vy(t) = Enter the equation for the total velocity versus time, v(t) = Enter the equation for the angle of the velocity vector versus time, θ(t)= Enter the maximum height of the football Enter the range of the football Enter the time it takes for the football to reach its maximum…arrow_forwardParticle A moves along the line y = 30 m with a constant velocity vector v of magnitude 3.0 m/s and parallel to the x axis. At the instant particle A passes the y axis , particle B leaves the origin with a zero initial speed and a constant acceleration vector a of magnitude 0.40 m/s2 . What angle theta between vector a and the positive direction of the y axis would result in a collision?arrow_forward
- On a spacecraft two engines fire for a time of 389 s. One gives the craft an acceleration in the x direction of ax = 3.41 m/s^2, while the other produces an acceleration in the y direction of ay = 7.34 m/s^2. At the end of the firing period, the craft has velocity components of vx = 1860 m/s and vy = 4290 m/s. Find the (a) magnitude and (b) direction of the initial velocity. Express the direction as an angle with respect to the +x axis.arrow_forwardIf a spaceship traveling through deep space with the velocity vector as a function of time: v = (2t i +4t j+ 6t k) m/s. (time is in seconds). Given the initial position of the spaceship is r0= (20 i− 50 j − 100 k ) m. At what time would the spaceship be the closest to the origin? and What would the acceleration of the spaceship be at this instant? If you can express the acceleration in Cartesian vector form. Also ? is the unit vector in the z-direction.arrow_forwardA particle moves along xy-plane with a position of x (t) = sin(2t) + 3 cos(4t) y (t) = 1/2 sin(t) – 6 cos(3t) a. Determine the position vector of the particle. b. Determine the velocity vector of the particle. c. Determine the acceleration vector of the particle. d. Determine the position vector of the particle when t = 4s. e. Determine the average acceleration from t = 0s to t= 5s. f. Determine the instantaneous acceleration when t= 1s. g. Determine the average velocity from t = 1s to t= 3s.arrow_forward
- A particle moves in the xy plane, starting from the origin at t = 0 with an initial velocity having an x component of 20 m/s and a y component of 215 m/s. The particle experiences an acceleration in the x direction, given by ax =4.0 m/s2.(A) Determine the total velocity vector at any later time. (B) Calculate the velocity and speed of the particle at t = 5.0 s and the angle the velocity vector makes with the x axis. (C) Determine the x and y coordinates of the particle at any time t and its position vector at this time.arrow_forwardA particle starts from the origin of a three-dimensional coordinate system and undergoes two consecutive displacements: r1→=2.0miˆ+1.0mjˆ+3.0mkˆr2→=−1.0miˆ−3.0mjˆ−1.0mkˆr1→=2.0mi^+1.0mj^+3.0mk^r2→=−1.0mi^−3.0mj^−1.0mk^ What is the distance of the particle from the origin after these two displacements?arrow_forwardConsider the motion of a bullet that is fired from a rifle 3 m above the ground in a northeast direction. The initial velocity of the bullet is150,150,0. Assume the x-axis points east, the y-axis points north, the positive z-axis is vertical (opposite g), the ground is horizontal, and only the gravitational force acts on the object. a. Find the velocity and position vectors for t≥0. b. Make a sketch of the trajectory. c. Determine the time of flight and range of the bullet. d. Determine the maximum height of the bullet.arrow_forward
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