Differentiate the trajectory Equation 3.14 to find its slope, tan u = dy/dx, and show

Under what conditions is the magnitude of the vector sumAS+ BS equal to the sum of the magnitudes of the two vectors?

Can two vectors of equal magnitude sum to zero? How about two vectors of unequal magnitude?

Repeat Question 2 for three vectors.

Can an object have a southward acceleration while moving northward? A westward acceleration while moving northward?

Youre a passenger in a car rounding a curve. The driver claims the car isnt accelerating because the speedometer reading is unchanging. Explain why the driver is wrong.

In what sense is Equation 3.8 really two (or three) equations?

Is a projectiles speed constant throughout its parabolic trajectory?

Is there any point on a projectiles trajectory where velocity and acceleration are perpendicular?

How is it possible for an object to be moving in one direction but accelerating in another?

Youre in a bus moving with constant velocity on a level road when you throw a ball straight up. When the ball returns, does it land ahead of you, behind you, or back at your hand? Explain.

Which of the following are legitimate mathematical equations? Explain. (a) v = 5i n m/s; (b) v!= 5 m/s; (c) a!= dv/dt;(d) a!= dv!/dt; (e) v!= 5in m/s.

You would probably reject as unscientific any claim that Earth is flat. Yet the assumption of Section 3.5 that leads to parabolic projectile trajectories is tantamount to assuming a flat Earth. Explain.

You walk west 220 m, then north 150 m. What are the magnitude and direction of your displacement vector?

An ion in a mass spectrometer follows a semicircular path of radius 15.2 cm. What are (a) the distance it travels and (b) the magnitude of its displacement?

A migrating whale follows the west coast of Mexico and North America toward its summer home in Alaska. It first travels 360 km northwest to just off the coast of northern California, and then turns due north and travels 400 km toward its destination. Determine graphically the magnitude and direction of its displacement.

Vector A S has magnitude 3.0 m and points to the right; vector B S has magnitude 4.0 m and points vertically upward. Find the magnitude and direction of vector C S such that A S + B S + C S = 0 S .

Use unit vectors to express a displacement of 120 km at 29 counterclockwise from the x-axis.

Find the magnitude of the vector 34ni + 13nj m and determine its angle to the x-axis.

(a) Whats the magnitude of ni + nj? (b) What angle does it make with the x-axis?

Youre leading an international effort to save Earth from an asteroid heading toward us at 15 km/s. Your team mounts a rocket on the asteroid and fires it for 10 min, after which the asteroid is moving at 19 km/s at 28 to its original path. In a news conference, what do you report for the magnitude of the acceleration imparted to the asteroid?

An object is moving at 18 m/s at 220 counterclockwise from the x-axis. Find the x- and y-components of its velocity

A car drives north at 40 mi/h for 10 min, then turns east and goes 5.0 mi at 60 mi/h. Finally, it goes southwest at 30 mi/h for 6.0 min. Determine the cars (a) displacement and (b) average velocity for this trip.

An objects velocity is v ! = ct3ni + dnj, where t is time and c and d are positive constants with appropriate units. Whats the direction of the objects acceleration?

A car, initially going eastward, rounds a 90 curve and ends up heading southward. If the speedometer reading remains constant, whats the direction of the cars average acceleration vector?

What are (a) the average velocity and (b) the average acceleration of the tip of the 2.4-cm-long hour hand of a clock in the interval from noon to 6 pm? Use unit vector notation, with the x-axis pointing toward 3 and the y-axis toward noon.

An ice skater is gliding along at 2.4 m/s, when she undergoes an acceleration of magnitude 1.1 m/s2 for 3.0 s. After that shes moving at 5.7 m/s. Find the angle between her acceleration vector and her initial velocity. Hint: You dont need to do a complicated calculation.

An object is moving in the x-direction at 1.3 m/s when it undergoes an acceleration a ! = 0.52nj m/s2 . Find its velocity vector after 4.4 s.

Youre a pilot beginning a 1500-km flight. Your planes speed is 1000 km/h, and air traffic control says youll have to head 15 west of south to maintain a southward course. If the flight takes 100 min, whats the wind velocity?

You wish to row straight across a 63-m-wide river. You can row at a steady 1.3 m/s relative to the water, and the river flows at 0.57 m/s. (a) What direction should you head? (b) How long will it take you to cross the river?

A plane with airspeed 370 km/h flies perpendicularly across the jet stream, its nose pointed into the jet stream at 32 from the perpendicular direction of its flight. Find the speed of the jet stream.

A flock of geese is attempting to migrate due south, but the wind is blowing from the west at 5.1 m/s. If the birds can fly at 7.5 m/s relative to the air, what direction should they head?

The position of an object as a function of time is given by r ! = 13.2t + 1.8t 2 2ni + 11.7t - 2.4t 2 2nj m, with t in seconds. Find the objects acceleration vector

Youre sailboarding at 6.5 m/s when a wind gust hits, lasting 6.3 s accelerating your board at 0.48 m/s2 at 35 to your original direction. Find the magnitude and direction of your displacement during the gust.

You toss an apple horizontally at 8.7 m/s from a height of 2.6 m. Simultaneously, you drop a peach from the same height. How long does each take to reach the ground?

A carpenter tosses a shingle horizontally off an 8.8-m-high roof at 11 m/s. (a) How long does it take the shingle to reach the ground? (b) How far does it move horizontally?

An arrow fired horizontally at 41 m/s travels 23 m horizontally. From what height was it fired?

Droplets in an ink-jet printer are ejected horizontally at 12 m/s and travel a horizontal distance of 1.0 mm to the paper. How far do they fall in this interval?

Protons drop 1.2 m over the 1.7-km length of a particle accelerator. Whats their approximate average speed?

If you can hit a golf ball 180 m on Earth, how far can you hit it on the Moon? (Your answer will be an underestimate because it neglects air resistance on Earth.)

Chinas high-speed rail network calls for a minimum turn radius of 7.0 km for 350-km/h trains. Whats the magnitude of a trains acceleration in this case?

The minute hand of a clock is 7.50 cm long. Find the magnitude of the acceleration of its tip.

How fast would a car have to round a 75-m-radius turn for its acceleration to be numerically equal to that of gravity?

Estimate the acceleration of the Moon, which completes a nearly circular orbit of 384.4 Mm radius in 27 days.

Global Positioning System (GPS) satellites circle Earth at altitudes of approximately 20,000 km, where the gravitational acceleration has 5.8% of its surface value. To the nearest hour, whats the orbital period of the GPS satellites?

Two vectors A S and B S have the same magnitude A and are at right angles. Find the magnitudes of (a) A S + 2B S and (b) 3A S - B S .

Vector A S has magnitude 1.0 m and points 35 clockwise from the x-axis. Vector B S has magnitude 1.8 m. Find the direction of B S such that A S + B S is in the y-direction.

Let A S = 15ni - 40nj and B S = 31nj + 18k n. Find C S such that A S + B S + C S = 0 S .

A biologist looking through a microscope sees a bacterium at r ! 1 = 2.2ni + 3.7nj - 1.2kn m. After 6.2 s, its located at r ! 2 = 4.6ni + 1.9k n m. Find (a) its average velocity, expressed in unit vectors, and (b) its average speed.

A particles position is r ! = 1ct2 - 2dt3 2ni + 12ct2 - dt3 2nj, where c and d are positive constants. Find expressions for times t 7 0 when the particle is moving in (a) the x-direction and (b) the y-direction.

For the particle in Problem 49, is there any time t 7 0 when the particle is (a) at rest and (b) accelerating in the x-direction? If either answer is yes, find the time(s).

Youre designing a cloverleaf highway interchange. Vehicles will exit the highway and slow to a constant 70 km/h before negotiating a circular turn. If a vehicles acceleration is not to exceed 0.40g (i.e., 40% of Earths gravitational acceleration), then whats the minimum radius for the turn? Assume the road is flat, not banked (more on this in Chapter 5).

An object undergoes acceleration 2.3ni + 3.6nj m/s2 for 10 s. At the end of this time, its velocity is 33ni + 15nj m/s. (a) What was its velocity at the beginning of the 10-s interval? (b) By how much did its speed change? (c) By how much did its direction change? (d) Show that the speed change is not given by the magnitude of the acceleration multiplied by the time. Why not?

The New York Wheel is the worlds largest Ferris wheel. Its 183 meters in diameter and rotates once every 37.3 min. Find the magnitudes of (a) the average velocity and (b) the average acceleration at the wheels rim, over a 5.00-min interval. (c) Compare your answer to (b) with the wheels instantaneous accelerations.

A ferryboat sails between towns directly opposite each other on a river, moving at speed v relative to the water. (a) Find an expression for the angle it should head at if the river flows at speed V. (b) Whats the significance of your answer if V 7 v?

The sum of two vectors, A S + B S , is perpendicular to their difference, A S - B S . How do the vectors magnitudes compare?

Write an expression for a unit vector at 45 clockwise from the x-axis

An object is initially moving in the x-direction at 4.5 m/s, when it undergoes an acceleration in the y-direction for a period of 18 s. If the object moves equal distances in the x- and y-directions during this time, whats the magnitude of its acceleration?

A particle leaves the origin with its initial velocity given by v ! 0 = 11ni + 14nj m/s, undergoing constant acceleration a ! = -1.2ni + 0.26nj m/s2 . (a) When does the particle cross the y-axis? (b) Whats its y-coordinate at the time? (c) How fast is it moving, and in what direction?

A kid fires a squirt gun horizontally from 1.6 m above the ground. It hits another kid 2.1 m away square in the back, 0.93 m above the ground. What was the waters initial speed?

A projectile has horizontal range R on level ground and reaches maximum height h. Find an expression for its initial speed.

You throw a baseball at a 45 angle to the horizontal, aiming at a friend whos sitting in a tree a distance h above level ground. At the instant you throw your ball, your friend drops another ball. (a) Show that the two balls will collide, no matter what your balls initial speed, provided its greater than some minimum value. (b) Find an expression for that minimum speed.

In a chase scene, a movie stuntman runs horizontally off the flat roof of one building and lands on another roof 1.9 m lower. If the gap between the buildings is 4.5 m wide, how fast must he run to cross the gap?

Standing on the ground 3.0 m from a building, you want to throw a package from your 1.5-m shoulder level to someone in a window 4.2 m above the ground. At what speed and angle should you throw the package so it just barely clears the windowsill?

Derive a general formula for the horizontal distance covered by a projectile launched horizontally at speed v0 from height h.

Consider two projectiles launched on level ground with the same speed, at angles 45 { a. Show that the ratio of their flight times is tan1a + 452.

You toss a protein bar to your hiking companion located 8.6 m up a 39 slope, as shown in Fig. 3.24. Determine the initial velocity vector so that when the bar reaches your friend, its moving horizontally

of position y versus x to determine the nature of the objects path. Then determine the magnitudes of the objects velocity and acceleration. Time, t (s) x (m) y (m) Time, t (s) x (m) y (m) 0 0 0 0.70 2.41 3.15 0.10 0.65 0.09 0.80 2.17 3.75 0.20 1.25 0.33 0.90 1.77 4.27 0.30 1.77 0.73 1.00 1.25 4.67 0.40 2.17 1.25 1.10 0.65 4.91 0.50 2.41 1.85 1.20 0.00 5.00 0.60 2.50 2.50

The table below lists position versus time for an object moving DATA in the xy plane, which is horizontal in this case. Make a plot A projectile launched at angle u to the horizontal reaches maximum height h. Show that its horizontal range is 4h>tan u.

As an expert witness, youre testifying in a case involving a motorcycle accident. A motorcyclist driving in a 60-km/h zone hit a stopped car on a level road. The motorcyclist was thrown from his bike and landed 39 m down the road. Youre asked whether he was speeding. Whats your answer?

Show that, for a given initial speed, the horizontal range of a projectile is the same for launch angles 45 + a and 45 - a.

A basketball player is 15 ft horizontally from the center of the basket, which is 10 ft off the ground. At what angle should the player aim the ball from a height of 8.2 ft with a speed of 26 ft/s?

Two projectiles are launched simultaneously from the same point, with different launch speeds and angles. Show that no combination of speeds and angles will permit them to land simultaneously and at the same point.

Consider the two projectiles in GOT IT? 3.5. Suppose the 45 projectile is launched with speed v and that its in the air for time t. Find expressions for (a) the launch speed and (b) the flight time of the 60 projectile, in terms of v and t.

The portion of a projectiles parabolic trajectory in the vicinity of the peak can be approximated as a circle. If the projectiles speed at the peak of the trajectory is v, formulate an argument to show that the curvature radius of the circle that approximates the parabola is r = v2 /g

A jet is diving vertically downward at 1200 km/h. If the pilot can withstand a maximum acceleration of 5g (i.e., 5 times Earths gravitational acceleration) before losing consciousness, at what height must the plane start a 90 circular turn, from vertical to horizontal, in order to pull out of the dive? See Fig. 3.25, assume the speed remains constant, and neglect gravity.

Your alpine rescue team is using a slingshot to send an emergency medical packet to climbers stranded on a ledge, as shown in Fig. 3.26; your job is to calculate the launch speed. What do you report?

If you can throw a stone straight up to height h, whats the maximum horizontal distance you could throw it over level ground?

In a conversion from military to peacetime use, a missile with maximum horizontal range 180 km is being adapted for studying Earths upper atmosphere. What is the maximum altitude it can achieve if launched vertically?

A soccer player can kick the ball 28 m on level ground, with its initial velocity at 40 to the horizontal. At the same initial speed and angle to the horizontal, what horizontal distance can the player kick the ball on a 15 upward slope?

A diver leaves a 3-m board on a trajectory that takes her 2.5 m above the board and then into the water 2.8 m horizontally from the end of the board. At what speed and angle did she leave the board?

Using calculus, you can find a functions maximum or minimum by differentiating and setting the result to zero. Do this for Equation 3.15, differentiating with respect to u, and thus verify that the maximum range occurs for u = 45.

Youre a consulting engineer specializing in athletic facilities, and youve been asked to help design the Olympic ski jump pictured in Fig. 3.27. Skiers will leave the jump at 28 m/s and 9.5 below the horizontal, and land 55 m horizontally from the end of the jump. Your job is to specify the slope of the ground so skiers trajectories make an angle of only 3.0 with the ground on landing, ensuring their safety. What slope do you specify?

Differentiate the trajectory Equation 3.14 to find its slope, tan u = dy/dx, and show that the slope is in the direction of the projectiles velocity, as given by Equations 3.10 and 3.11.

36 m/s. Some members of the class think a 45 launch angle will give the maximum range, but others claim the cliff height makes a difference. What do you give for the angle that will maximize the range?

Your medieval history class is constructing a trebuchet, a catapult-like weapon for hurling stones at enemy castles. The plan is to launch stones off a 75-m-high cliff, with initial speed Generalize Problem 84 to find an expression for the angle that will maximize the range of a projectile launched with speed v0 from height h above level ground.

(a) Show that the position of a particle on a circle of radius R with its center at the origin is r ! = R1cos uni + sin unj2, where u is the angle the position vector makes with the x-axis. (b) If the particle moves with constant speed v starting on the x-axis at t = 0, find an expression for u in terms of time t and the period T to complete a full circle. (c) Differentiate the position vector twice with respect to time to find the acceleration, and show that its magnitude is given by Equation 3.16 and its direction is toward the center of the circle.

In dealing with nonuniform circular motion, as shown in Fig. 3.23, we should write Equation 3.16 as ar = v2 /r, to show that this is only the radial component of the acceleration. Recognizing that v is the objects speed, which changes only in the presence of tangential acceleration, differentiate this equation with respect to time to find a relation between the magnitude of the tangential acceleration and the rate of change of the magnitude of the radial acceleration. Assume the radius stays constant.

Repeat Problem 87, now generalizing to the case where not only the speed but also the radius may be changing.

Which statement characterizes the distances the students travel? a. Theyre equal. b. C 7 A 7 B c. C 7 B 7 A d. B 7 C 7 A

Alice (A), Bob (B), and Carrie (C) all start from their dorm and head for the library for an evening study session. Alice takes a straight path, while the paths Bob and Carrie follow are portions of circular arcs, as shown in Fig. 3.28. Each student walks at a constant speed. All three leave the dorm at the same time, and they arrive simultaneously at the library Which statement characterizes the students displacements? a. Theyre equal. b. C 7 A 7 B c. C 7 B 7 A d. B 7 C 7 A

Alice (A), Bob (B), and Carrie (C) all start from their dorm and head for the library for an evening study session. Alice takes a straight path, while the paths Bob and Carrie follow are portions of circular arcs, as shown in Fig. 3.28. Each student walks at a constant speed. All three leave the dorm at the same time, and they arrive simultaneously at the library Which statement characterizes their average speeds? a. Theyre equal. b. C 7 A 7 B c. C 7 B 7 A d. B 7 C 7 A

Alice (A), Bob (B), and Carrie (C) all start from their dorm and head for the library for an evening study session. Alice takes a straight path, while the paths Bob and Carrie follow are portions of circular arcs, as shown in Fig. 3.28. Each student walks at a constant speed. All three leave the dorm at the same time, and they arrive simultaneously at the library Which statement characterizes their accelerations while walking (not starting and stopping)? a. Theyre equal. b. None accelerates. c. A 7 B 7 C d. C 7 B 7 A e. B 7 C 7 A f. Theres not enough information to decide.