# Velocity and Pic

Review Example problems #1 - 12 by yourself.

Problem 3 (page 96): A web page designer creates an animation in which a dot on a computer screen has a position of r = [4 cm + (2.5 cm/s2)t2]i + (5 cm/s)t j. a) Find the magnitude and direction of the dot’s average velocity between t = 0 and t = 2 s. b) Find the magnitude and direction of the instantaneous velocity at t = 0, t = 1 s, nd t = 2 s. c) Sketch the dot’s trajectory from t = 0 to t = 2 s, and show the velocities calculated in part (b).

(a) Identify and Set Up: From [pic] we can calculate x and y for any t. Then use Eq. (3.2), in component form.

Execute: [pic]

At [pic] [pic]

At [pic] [pic]

[pic]

[pic]

| |[pic] | |[pic] | | | | |[pic] | | | | |[pic] | |Figure 3.3a | | |

Evaluate: Both x and y increase, so [pic] is in the 1st quadrant. (b) Identify and Set Up: Calculate [pic] by taking the time derivative of [pic] Execute: [pic]

[pic] [pic] [pic] [pic] and [pic]

[pic] [pic] [pic] [pic] and [pic]

[pic] [pic] [pic] [pic] and [pic]

(c) The trajectory is a graph of y versus x.

[pic] [pic]

For values of t between 0 and 2.0 s, calculate x and y and plot y versus x.

|[pic] | |Figure 3.3b |

Evaluate: The sketch shows that the instantaneous velocity at any t is tangent to the trajectory.

Problem 5 (page 96): A jet plane is flying at a constant altitude. At time t1 = 0 it has components of velocity vx = 90 m/s, vy = 110 m/s. At time t2 = 30 s the components are vx = -170 m/s, vy = 40 m/s. a) Sketch the velocity vectors at t1 and t2.

b) For this time interval calculate the components of the average acceleration, c) The magnitude and direction of the average acceleration.

Identify and Set Up: Use Eq. (3.8) in component form to calculate [pic] and [pic] Execute: (a) The velocity vectors at [pic] and [pic] are shown in Figure 3.5a.

|[pic] | |Figure 3.5a |

(b) [pic]

[pic]

|(c)|[pic] | |[pic] | | | | |[pic] | | | | |[pic] | |Figure 3.5b | | |

Evaluate: The changes in [pic] and [pic] are both in the negative x or y direction, so both components of [pic] are in the 3rd quadrant.

Problem 16 (page 97): On level ground a shell is fired with an initial velocity of 50 m/s at 60o above the horizontal and feels no appreciable air resistance. a) Find the horizontal and vertical components of the shell’s initial velocity. b) How long does it take the shell to reach its highest point? c) Find its maximum height above the ground.

d) How far from its firing point does the shell land?

e) At its highest point, find the horizontal and vertical components of its acceleration and velocity....

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