Chapter 3

A squirrel has \(x\)- and \(y\)-coordinates (1.1 m, 3.4 m) at time \(t_1\) = 0 and coordinates (5.3 m, -0.5 m) at time \(t_2\) = 3.0 s. For this time interval, find (a) the components of the average velocity, and (b) the magnitude and direction of the average velocity.

A rhinoceros is at the origin of coordinates at time \(t_1\) = 0. For the time interval from \(t_1\) = 0 to \(t_2\) = 12.0 s, the rhino's average velocity has \(x\)-component -3.8 m/s and y-component 4.9 m/s. At time \(t_2\) = 12.0 s, (a) what are the \(x\)- and \(y\)-coordinates of the rhino? (b) How far is the rhino from the origin?

CALC A web page designer creates an animation in which a dot on a computer screen has position $$ \vec{r} =[34.0 cm +(2.5 cm/s^2)t^2] \hat{i} +(5.0 cm/s)t \hat{\jmath}.$$ (a) Find the magnitude and direction of the dot's average velocity between \(t\) = 0 and \(t\) = 2.0 s.(b) Find the magnitude and direction of the instantaneous velocity at \(t\) = 0, \(t\) = 1.0 s, and \(t\) = 2.0 s. (c) Sketch the dot's trajectory from \(t\) = 0 to \(t\) = 2.0 s, and show the velocities calculated in part (b).

CALC The position of a squirrel running in a park is given by \(\vec{r}= [(0.280 m/s)t + (0.0360 m/s^2)t^2] \hat{\imath}+(0.0190 m/s^3)t^3\hat{\jmath}\). (a) What are \(v_x(t)\) and \(v_y(t)\), the \(x\)- and \(y\)-components of the velocity of the squirrel, as functions of time? (b) At \(t\) = 5.00 s, how far is the squirrel from its initial position? (c) At \(t\) = 5.00 s, what are the magnitude and direction of the squirrel's velocity?

A jet plane is flying at a constant altitude. At time \(t_1\) = 0, it has components of velocity \(v_x\) = 90 m/s, \(v_y\) = 110 m/s. At time \(t_2\) = 30.0 s, the components are \(v_x\) = -170 m/s, \(v_y\) = 40 m/s. (a) Sketch the velocity vectors at \(t_1\) and \(t_2\). How do these two vectors differ? For this time interval calculate (b) the components of the average acceleration, and (c) the magnitude and direction of the average acceleration.

A dog running in an open field has components of velocity \(v_x\) = 2.6 m/s and \(v_y\) = -1.8 m/s at \(t_1\) = 10.0 s. For the time interval from \(t_1\) = 10.0 s to \(t_2\) = 20.0 s, the average acceleration of the dog has magnitude 0.45 m/s\(^2\) and direction 31.0\(^\circ\) measured from the +\(x\)-axis toward the +\(y\)-axis. At \(t_2\) = 20.0 s, (a) what are the \(x\)- and \(y\)-components of the dog's velocity? (b) What are the magnitude and direction of the dog's velocity? (c) Sketch the velocity vectors at \(t_1\) and \(t_2\). How do these two vectors differ?

CALC The coordinates of a bird flying in the \(xy\)-plane are given by \(x(t)\) = \(at\) and \(y(t)\) = 3.0 m - \(\beta t^2\), where \(\alpha\) = 2.4 m/s and \(\beta\) = 1.2 m/s\(^2\). (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?

CALC A remote-controlled car is moving in a vacant parking lot. The velocity of the car as a function of time is given by \(\hat{v} =[5.00 m/s - (0.0180 m/s^3)t^2] \hat{\imath} = [2.00 m/s + (0.550 m/s^2)t]\hat{j}\). (a) What are \(a_x(t)\) and \(a_y(t)\), the \(x\)- and \(y\)-components of the car's velocity as functions of time? (b) What are the magnitude and direction of the car's velocity at \(t\) = 8.00 s? (b) What are the magnitude and direction of the car's acceleration at \(t\) = 8.00 s?

A physics book slides off a horizontal tabletop with a speed of 1.10 m/s. It strikes the floor in 0.480 s. Ignore air resistance. Find (a) the height of the tabletop above the floor; (b) the horizontal distance from the edge of the table to the point where the book strikes the floor; (c) the horizontal and vertical components of the book's velocity, and the magnitude and direction of its velocity, just before the book reaches the floor. (d) Draw \(x-t, y-t, v_x-t\), and \(v_y-t\) graphs for the motion.

Crickets Chirpy and Milada jump from the top of a vertical cliff. Chirpy drops downward and reaches the ground in 2.70 s, while Milada jumps horizontally with an initial speed of 95.0 cm/s. How far from the base of the cliff will Milada hit the ground? Ignore air resistance.