Physics: Exam 1

To find the distance traveled by the light beam without using the Pythagorean theorem, you would multiply the distance of 46.0 m by:

1 / cos 39.0°

Suppose the river is flowing east at 3 m/s and the boat is traveling south at 4 m/s with respect to the river. Find the speed and direction of the boat relative to Earth.

Vbe= √x^2 + y^2 θ = tan−1 (x/y)

A ball is thrown so that its initial vertical and horizontal components of velocity are 40 m/s and 20 m/s, respectively. Use a motion diagram to estimate the ball's total time of flight and the distance it traverses before hitting the ground.

Vy= V-V t= (V-Vo) / -g ∆x= Vx t

A high fountain of water is located at the center of a circular pool as shown in the figure below. Not wishing to get his feet wet, a student walks around the pool and measures its circumference to be 16.0 m. Next, the student stands at the edge of the pool and uses a protractor to gauge the angle of elevation at the bottom of the fountain to be 55.0°. How high is the fountain?

r= c/ 2π height= r tan θ

Suppose your hair grows at the rate of 1/72 inch per day. Find the rate at which it grows in nanometers per second. Because the distance between atoms in a molecule is on the order of 0.1 nm, your answer suggests how rapidly atoms are assembled in this protein synthesis.

rate = (1 in /72 day) x (1 day / 24 h) x (1 h / 3600 s) x (2.54 cm / 1.00 in) x (10^9 nm / 10^2 cm) = 4.08 nm/s

The left side of an equation has dimensions of length and the right side has dimensions of length squared. Can the equation be correct?

No, because the equation is dimensionally inconsistent.

A map suggests that Atlanta is 730 miles in a direction 5.00° north of east from Dallas. The same map shows that Chicago is 560 miles in a direction 21.0° west of north from Atlanta. The figure below shows the location of these three cities. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago.

Rx= (730mi)cos5.00˚ - (560 mi)sin21.0˚= 527 mi Ry= (730mi)sin5.00˚ - (560 mi)cos21.0˚= 586 mi R= √(Rx^2 + Ry^2)= 788mi θ=tan^-1 (Ry/Rx)

How does the magnitude of the velocity vector at impact compare with the magnitude of the initial velocity vector?

They are the same since the magnitude of the vertical component of velocity is the same at each height on the way up and on the way down.

Neglecting air friction effects, what path does the package travel as observed by the pilot?

a vertical line downward

As a projectile moves in its parabolic path, the velocity and acceleration vectors are perpendicular to each other at which of the following along the projectile's path?

at the peak

A ball is thrown so that its initial vertical and horizontal components of velocity are 60 m/s and 15 m/s, respectively. Estimate the maximum height the ball reaches.

∆x= Voy/2g

The best leaper in the animal kingdom is the puma, which can jump to a height of 3.7 m when leaving the ground at an angle of 45°. With what speed must the animal leave the ground to reach that height?

Voy= √2g∆y Vo= Voy / sinθ

A baseball player moves in a straight-line path in order to catch a fly ball hit to the outfield. His velocity as a function of time is shown in figure. What is his instantaneous acceleration at point C?

-2 m/s2

Can a tangent line to a velocity vs. time graph ever be vertical? Explain.

-It would correspond to an infinite instantaneous acceleration -No

List some advantages of the metric system of units over most other systems of units.

-The metric system uses a single base unit of measure for each physical quantity. -Metric measurements are set by international agreement and are the same in every country.

A rectangular airstrip measures 30.90 m by 290 m, with the width measured more accurately than the length. Find the area (in m2), taking into account significant figures.

A= LW = (290 m)(30.90 m) (rounded to two significant figures)

A commuter airplane starts from an airport and takes the route shown in the figure below. The plane first flies to city A, located 175 km away in a direction 30.0° north of east. Next, it flies for 150 km 20.0° west of north, to city B. Finally, the plane flies 190 km due west, to city C. Find the location of city C relative to the location of the starting point.

Ax= a cosθ Bx= b cosθ Cx= c cosθ Ay= a sinθ By= b sin θ Cy= c sin θ Rx=Ax+Bx+Cx Ry=Ay+By+Cy R= √(Rx^2 + Ry^2) θ=tan^-1 (Rx/Ry)

A baseball is thrown from the outfield toward the catcher. When the ball reaches its highest point, which statement is true?

Its velocity is perpendicular to its acceleration.

A baseball player moves in a straight-line path in order to catch a fly ball hit to the outfield. His velocity as a function of time is shown in figure (a). Find his instantaneous acceleration at points circled A, circled B, and circled C.

a=∆v/∆t

A tennis player on serve tosses a ball straight up. As the tennis ball travels through the air, its speed does which of the following?

decreases and then increases

Starting with the answers to part (b), work backwards to recover the given radius and angle. Why are there slight differences from the original quantities?

rounding the final calculated values of x and y in the example before using them to work backwards

(a) If a person can jump a maximum horizontal distance (by using a 45° projection angle) of 1.41 m on Earth, what would be his maximum range on the Moon, where the free-fall acceleration is g/6 and g = 9.80 m/s2? (b) Repeat for Mars, where the acceleration due to gravity is 0.38g.

(a) 6∆x (b) ∆x/0.38

True or False? (a) A car must always have an acceleration in the same direction as its velocity. (b) It's possible for a slowing car to have a positive acceleration. (c) An object with constant nonzero acceleration can never stop and remain at rest.

(a) False (b) True (c) True

How would the time of the jump and the horizontal distance traveled change if g were changed, for example if the jump could be repeated with the same initial velocity on a different planet?

-The time of the jump increases when g is smaller. -The displacement increases with increased time of the jump.

How would the answer to part (b), the maximum height, change if the person throwing the ball jumped upward at the instant he released the ball?

The maximum height would increase.

Suppose you are carrying a ball and running at a constant speed, and wish to throw the ball and catch it as it comes back down. Neglecting air resistance, you should do which of the following?

Throw the ball straight up in the air and maintain the same speed.

Repeat the conversion, using the relationship 1.00 m/s = 2.24 mi/h. Why is the answer slightly different?

Using the conversion factor fails to keep extra digits until the final answer.

The speed of light is about 3.00 ✕ 10^8 m/s. Convert this figure to miles per hour.

c = (3.00 ✕ 10^8 m/s) x (3600 s / 1 h) x (1 km / 10^3m) x (1 mi / 1.609 km) = 6.71 ✕ 108 mi/h.

trig functions

x=rcosθ y=rsinθ

A farm truck moves due east with a constant velocity of 8.00 m/s on a limitless, horizontal stretch of road. A boy riding on the back of the truck throws a can of soda upward (see figure below) and catches the projectile at the same location on the truck bed, but 12.0 m farther down the road. (a) In the frame of reference of the truck, at what angle to the vertical does the boy throw the can? (b) What is the initial speed of the can relative to the truck? (c) What is the shape of the can's trajectory as seen by the boy? An observer on the ground watches the boy throw the can and catch it. (d) In this observer's frame of reference, describe the shape of the can's path. (e) In this observer's frame of reference, determine the initial velocity of the can.

(a) 0˚ (because the can returns to the same spot on the truck bed, which gives the can zero horizontal velocity) (b) tup= 1/2 (∆x/Vx) Voy= 0 + at (c) a straight line segment upward and then downward (d) a symmetric section of a parabola opening downward (e) Vo = √(Vox2 + Voy2) θ = tan−1 (Vox/Voy)

Estimate the time duration of each of the following. (a) a heartbeat (Enter your answer in s.) (b) a football game (Enter your answer in hours.) (c) a summer (Enter your answer in months.) (d) a movie (Enter your answer in hours.) (e) the blink of an eye (Enter your answer in s.)

(a) 1s (b) 3 hr (c) 3 months (d) 2 hours (e) 0.3 s

A hiker begins a trip by first walking 25.0 km 45.0° south of east from her base camp. On the second day she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger's tower. (a) Determine the components of the hiker's displacements in the first and second days. (b) Determine the components of the hiker's total displacement for the trip. (c) Find the magnitude and direction of the displacement from base camp

(a) Ax = A cos (-45.0°) Ay = A sin -(45.0°) Bx = B cos 60.0° By = B sin 60.0° (b) Rx = Ax + Bx Ry = Ay + By (c) R = √(Rx^2 + Ry^2) θ=tan^-1 (Ry/Rx)

A paper in the journal Current Biology tells of some jellyfish-like animals that attack their prey by launching stinging cells in one of the animal kingdom's fastest movements. High-speed photography showed the cells were accelerated from rest for 700 ns at 5.30 ✕ 107 m/s2. Calculate the maximum speed reached by the cells and the distance traveled during the acceleration. (a) Calculate the maximum speed (in m/s) reached by the cells (b) Calculate the distance (in m) traveled during the acceleration.

(a) V= Vo + at (b) ∆x= Vot + 1/2at^2 when Vo=0 ∆X= 1/2at^2

A truck covers 40.0 m in 8.60 s while uniformly slowing down to a final velocity of 3.00 m/s. (a) Find the truck's original speed. (b) Find its acceleration

(a) Vo= (∆x/t)2 - Vf (b) a= (Vf-Vo) / ∆t

A grasshopper jumps a horizontal distance of 1.10 m from rest, with an initial velocity at a 47.0° angle with respect to the horizontal. (a) Find the initial speed of the grasshopper. (b) Find the maximum height reached.

(a) Vo= √(∆xg) / 2cosθsinθ (b) ymax= (V^2sin^2θ) / 2g

A tennis player tosses a tennis ball straight up and then catches it after 1.88 s at the same height as the point of release. (a) What is the acceleration of the ball while it is in flight? (b) What is the velocity of the ball when it reaches its maximum height? (c) Find the initial velocity of the ball. (d) Find the maximum height it reaches.

(a) a= 9.8 m/s^2 downward (b) V max height=0 (has no direction) (C) ∆y=0 so, 0=Vot - 1/2gt^2 (d) ∆y max= (V^2 - Vo^2) / 2(-a)

A certain aircraft has a liftoff speed of 130 km/h. (a) What minimum constant acceleration does the aircraft require if it is to be airborne after a takeoff run of 213 m? (b) How long does it take the aircraft to become airborne?

(a) a= [v-0 (10^3 / 3600 s)]^2 / (2∆x) (b) ∆t= [v-0 (10^3 / 3600 s)] / a

Suppose the driver in this example is now moving with speed 36.0 m/s, and slams on the brakes, stopping the car in 4.2 s. (a) Find the acceleration assuming the acceleration is constant. (b) Find the distance the car travels, assuming the acceleration is constant. (c) Find the average velocity.

(a) a=∆v/∆t (b) x= Vot + 1/2 at^2 (c) V avg= ∆x/∆t

Carry out the following arithmetic operations. (Enter your answers to the correct number of significant figures.) (a) the sum of the measured values 521, 38.2, 0.90, and 9.0 (b) the product 0.0052 ✕ 469.9 (c) the product 19.10 ✕ π

(a) for summation and subtractions, your answer should have sig figs equal to the number with the least decimal places (b) for multiplication and division your answer should have equal to that of the number with the least sig figs (c) same as b

(a) The Cartesian coordinates of a point in the xy-plane are (x, y) = (−3.50 m, −2.50 m), as shown in the figure. Find the polar coordinates of this point. (b) Convert (r, θ) = (5.00 m, 37.0°) to rectangular coordinates.

(a) r = √(x^2 + y^2) θ=tan^-1 (y/x) *add 180˚ if in the 2 or 3 quadrant (b) x = rcos θ y = rsin θ

A jet plane lands with a speed of 96 m/s and can accelerate at a maximum rate of −4.50 m/s2 as it comes to rest. (a) From the instant the plane touches the runway, what is the minimum time needed before it can come to rest? (b) Can this plane land on a small tropical island airport where the runway is 0.800 km long?

(a) t= (V-Vo) / a (b) ∆X= (V-Vo / 2) x t

A projectile is launched straight up at 66 m/s from a height of 72.5 m, at the edge of a sheer cliff. The projectile falls, just missing the cliff and hitting the ground below. (a) Find the maximum height of the projectile above the point of firing. (b) Find the time it takes to hit the ground at the base of the cliff. (c) Find its velocity at impact.

(a) t= -Vo / -a ymax= vt - 1/2at^2 (b) -yo= Vot - 1/2at^2 (c) V= -at + Vo

A bartender slides a beer mug at 1.2 m/s towards a customer at the end of a frictionless bar that is 1.0 m tall. The customer makes a grab for the mug and misses, and the mug sails off the end of the bar. (a) How far away from the end of the bar does the mug hit the floor? (b) What are the speed and direction of the mug at impact?

(a) t= √2y/g x= Vxt (b) Vy= gt V=√(Vx^2 + Vy^2) θ= tan−1(Vy/Vx)

An Alaskan rescue plane drops a package of emergency rations to stranded hikers, as shown in the figure. The plane is traveling horizontally at 40.0 m/s at a height of 1.00 102 m above the ground. (a) Where does the package strike the ground relative to the point at which it was released? (b) What are the horizontal and vertical components of the velocity of the package just before it hits the ground? (c) Find the angle of the impact.

(a) t= √y/ (1/2g) x= Vox t (b) Vx= Vocosθ Vy= Vosinθ - gt (c) θ= tan−1(Vy/Vx)

A long jumper (see figure) leaves the ground at an angle of 20.0° to the horizontal and at a speed of 11.0 m/s. (a) How long does it take for her to reach maximum height? (b) What is the maximum height? (c) How far does she jump? (Assume her motion is equivalent to that of a particle, disregarding the motion of her arms and legs.) (d) Use the proper equation to find the maximum height she reaches.

(a) tmax= (Vo sinθ) / g (b) ymax= Vo sinθ)tmax - 1/2g(max)^2 (c) t= 2tmax ∆x= (Vo cosθ)t (d) ∆y= (Vy^2 - Vo^2) / -2g

(a) A race car starting from rest accelerates at a constant rate of 5.00 m/s^2. What is the velocity of the car after it has traveled 1.00 x10^2 ft? (b) How much time has elapsed? (c) Calculate the average velocity two different ways.

(a) v= √(vo^2 + 2a∆x) (b) v=at + vo--> when vo=0, t=v/a (c) V avg= (Xf-Xi)/(Tf-Ti) or V avg= Vo+V/2

A ball is thrown from the top of a building with an initial velocity of 20.0 m/s straight upward, at an initial height of 50.0 m above the ground. The ball just misses the edge of the roof on its way down, as shown in the figure. Determine (a) the time needed for the ball to reach its maximum height, (b) the maximum height, (c) the time needed for the ball to return to the height from which it was thrown and the velocity of the ball at that instant, (d) the time needed for the ball to reach the ground, and (e) the velocity and position of the ball at t = 5.00 s. Neglect air drag.

(a) v=at + Vo ∆y= y-yo=Vot + 1/2at^2 ---> t= -Vo/-9.8 (b) ymax= Vot - 1/2at^2 (c) plug in times for y=0 0= Vot - 1/2at^2 V= Vo + -at (d) y= Vot- 1/2at^2 (plug in times to find t) (e) v= -at + Vo y= Vot - 1/2at^2

The figure below shows the unusual path of a confused football player. After receiving a kickoff at his own goal, he runs downfield to within inches of a touchdown, then reverses direction and races back until he's tackled at the exact location where he first caught the ball. (a) the total distance he travels (b) his displacement (c) his average velocity in the x direction

(a) xi + xf (b) ∆x= xf - xi (c) v=∆x/t

A motorist drives north for 38.0 minutes at 81.0 km/h and then stops for 15.0 minutes. He then continues north, traveling 130 km in 2.20 h. (a) What is his total displacement? (b) What is his average velocity?

(a) ∆x= Vot(1h/60min) + x (b) ∆t= (t1 +t2) x (1h/60min) + tf Vavg= ∆x/∆t

A ball is thrown vertically upward with a speed of 32.0 m/s. (a) How high does it rise? (b) How long does it take to reach its highest point? (c) How long does the ball take to hit the ground after it reaches its highest point? (d) What is its velocity when it returns to the level from which it started?

(a) ∆y max= (V-Vo^2) / 2(-g) (b) tup= V-Vo/(-g) (c) t= √(2∆y)/g (d) v= 0 + (-g x t)

A person measures the height of a building by walking out a distance of 46.0 m from its base and shining a flashlight beam toward the top. When the beam is elevated at an angle of 39.0° with respect to the horizontal, as shown in the figure, the beam just strikes the top of the building. (a) If the flashlight is held at a height of 2.00 m, find the height of the building. (b) Calculate the length of the light beam.

(a) ∆y= tanθ x m (b) r = √(x^2 + y^2)

A stuntman sitting on a tree limb wishes to drop vertically onto a horse galloping under the tree. The constant speed of the horse is 14.0 m/s, and the man is initially 3.25 m above the level of the saddle. (a) What must be the horizontal distance between the saddle and limb when the man makes his move? (b) How long is he in the air?

(b) t= √(2 x ∆y) / a (a) ∆x= Vhorse x t

If a car is traveling at a speed of 28.0 m/s, is the driver exceeding the speed limit of 55.0 mi/h?

(mi x m) / (h x s)

A second hiker follows the same path the first day, but then walks 15.0 km east on the second day before turning and reaching the ranger's tower. How does the second hiker's resultant displacement vector compare with that of the first hiker? List all aspects that apply.

-The two displacements have the same direction. - The second hiker's displacement has the same magnitude as the first.

The figure below shows the unusual path of a confused football player. After receiving a kickoff at his own goal, he runs downfield to within inches of a touchdown, then reverses direction and races back until he's tackled at the exact location where he first caught the ball. During this run, what is the displacement of the player?

0 yd

A projectile is launched from Earth's surface at a certain initial velocity at an angle above the horizontal, reaching maximum height after time tmax. Another projectile is launched with the same initial velocity and angle from the surface of the Moon, where the acceleration of gravity is one-sixth that of Earth. Neglecting air resistance (on Earth), how long does it take the projectile on the Moon to reach its maximum height?

6 t max

A student throws a heavy red ball horizontally from a balcony of a tall building with an initial speed v0. At the same time, a second student drops a lighter blue ball from the same balcony. Neglecting air resistance, which statement is true?

The balls reach the ground at the same instant.

How long does it take an automobile traveling in the left lane of a highway at 50.0 km/h to overtake (become even with) another car that is traveling in the right lane at 20.0 km/h when the cars' front bumpers are initially 150 m apart?

Vab= (Vae - Vbe) x (1000/3600) ∆x= Vae+Vbe t= ∆x/Vab

The boat in the figure is heading due north as it crosses a wide river with a velocity of 10.0 km/h relative to the water. The river has a uniform velocity of 5.00 km/h due east. Determine the velocity of the boat with respect to an observer on the riverbank.

Vbe= √(Vx^2 + Vy^2) θ = tan−1 (Vx/Vy)

A typical jetliner lands at a speed of 1.60 x10^2 mi/h and decelerates at the rate of (10 mi/h)/s. If the plane travels at a constant speed of 1.60x10^2 mi/h for 1.00 s after landing before applying the brakes, what is the total displacement of the aircraft between touchdown on the runway and coming to rest?

Vo= v x (a/t) a= (-rate) x (a/t) ∆x coasting= Vo + 1/2(0)t^2 ∆x braking= (0-Vo^2)/(2xa) ∆x total= ∆x coasting+∆x braking

A tennis player on serve tosses a ball straight up. While the ball is in free fall, its acceleration does which of the following?

remains constant

While standing atop a building 49.3 m tall, you spot a friend standing on a street corner. Using a protractor and a dangling plumb bob, you find that the angle between the horizontal and the direction to the spot on the sidewalk where your friend is standing is 27°. Your eyes are located 1.78 m above the top of the building. How far away from the foot of the building is your friend?

total height= Y building + Y eyes b= height/tanθ

A jet lands at 62.7 m/s, the pilot applying the brakes 2.08 s after landing. Find the acceleration needed to stop the jet within 5.46 x10^2 m after touchdown.

x= Vt 546-x= ∆x a= -Vo^2/ (2 x ∆x)

If the speed of the boat relative to the water is increased, what happens to the angle?

θ decreases

The speed of a nerve impulse in the human body is about 100 m/s. If you accidentally stub you toe in the dark, estimate the time it takes the nerve impulse to travel to your brain. (Assume that you are approximately 1.90 m tall and that the nerve impulse travels at uniform speed.)

∆t=∆x/v

A brick is thrown upward from the top of a building at an angle of 15° to the horizontal and with an initial speed of 16 m/s. If the brick is in flight for 3.4 s, how tall is the building?

∆y= Voyt -1/2gt^2

distance (when given two points)

√(x2-x1)^2 + (y2-y1)^2)