CALC? The coordinates of a bird flying in the xy-plane are given by x(t) = ?t and y(t) = 3.0 m - ?t2, where ? = 2.4 m/s and ? = 1.2 m/s2. (a) Sketch the path of the bird between t = 0 and t = 2.0 s. (b) Calculate the velocity and acceleration vectors of the bird as functions of time. (c) Calculate the magnitude and direction of the bird’s velocity and acceleration at t = 2.0 s. (d) Sketch the velocity and acceleration vectors at t = 2.0 s. At this instant, is the bird’s speed increasing, decreasing, or not changing? Is the bird turning? If so, in what direction?

Solution 7E Step 1: Provided, x(t) = t y(t) = 3.0 m - t 2 2 = 2.4 m/s and = 1.2 m/s Put t = 0 s, then, x(0) = 2.4 m/s × 0 s = 0 m y(0) = 3.0m - (1.2 m/s × 0 s ) = 3.0m Put, t = 1 s, then, x(1) = 2.4 m/s × 1 s = 2.4 m 2 2 y(1) = 3.0m - (1.2 m/s × 1 s ) = 1.8 m Put, t = 2 s, then, x(2) = 2.4 m/s × 2 s = 4.8 m y(1) = 3.0m - (1.2 m/s × 4 s ) = - 1.8 m a) We can plot the graph as mentioned below in figure 1. The vertical axis represents y (t) and the horizontal axis represents x(t). Figure 1 Step 2: b) Velocity of the bird along x-direction, dx/dt, v = d/dt ( t) = x Velocity of the bird along y-direction, dy/dt, v = d/dt (3.0 m - t ) = d/dt (3 m) -d/dt (t ) 2 2 y dy/dt = v = y- 2t = - 2t 2 2 Acceleration of the bird along x-direction, d x/dt , a = d/dt () = 0 x 2 2 Acceleration of the bird along y-direction, d y/dt , a = d/dt (- 2t ) = - 2y Step 3: Velocity of the bird along x-direction when t = 2s,v = = 2.4 m/s x Velocity of the bird along y-direction when t = 2s, v = - 2t = - 2 × 1.2 m/s ×y2 s = - 4.8 2 m/s Magnitude of bird’s velocity, v = .4 + 4.8 = 5.76 + 23.04 = 28.08 = 5.366 m/s In order to find the direction, we should take the ratio of v / v y x vy v sin vx v cos Therefore, v /y xv sin /v cos = tan = - 4.8 m/s/3.0 m/s = - 1.6 = tan ( -1.6) = - 58°