A. Basic Concepts and Graphs A01 [Qual] [Easy] For each of the following, select if it is a vector or a scalar. a) Speed b) Distance traveled c) Velocity d) (Linear) Displacement A02 [Qual] [Easy] Give an example where distance traveled, displacement and the magnitude of the displacement are not the same. A03 [Qual] [Easy] What is the connection between instantaneous velocity and instantaneous speed? A04 [Qual] [extra practice] [Easy] Give an example where distance traveled, displacement and the magnitude of the displacement are the same. A05 [Easy] A bus travels from Montreal to Ottawa (distance 200. km) in 2.0 h, stays there for 1.0 hour and then travels back to Montreal in another 2.0 h. Considering the round trip, a) What is the average speed of the bus? b) What is the average velocity of the bus? A06 A v-t graph plots the velocity (v) of an object as a function of time (t). The x-axis represents the time, the y-axis represents the velocity of the object at each moment. In the example shown, an object has no velocity between t=0.0 s and 2.0 s. It then speeds up to +4.0 m/s at t=4.0 s and keeps that velocity until t=6.0 s. Between t=6.0 s and 10.0 s, it is slowing down to 0.0 m/s. a) Find the acceleration at t=1.0 s, 3.0 s, 5.0 s and 8.0 s. b) Between t=0.0 s and t=2.0 s, how much did the position change? c) Between t=2.0 s and t=4.0 s, how much did the position change?
d) Between t=4.0 s and t=6.0 s, how much did the position change? e) Where is the object at t=10.0 s if it started at s(t=0.0 s)=0. 0 m. A07 An s-t graph plots the position (s) of an object as a function of time (t). The x-axis represents the time, the y-axis represents the position of an object. In the example shown, an object has moved in 10.0 seconds from its initial position at 0.0 m to 20.0 m. a) What is the average velocity (magnitude and direction) between 0.0 and 2.0 seconds? b) What is the average velocity (magnitude and direction) between 4.0 and 6.0 seconds? c) What is the velocity at t=3.0 s and t=8.0 s? d) What is the average acceleration between 3.0s and 8.0s? e) (optional) Draw a v-t (velocity-time) Graph with time on the x-axis and v on the y-axis. Don't forget the units! A08 An a-t graph plots the acceleration (a) of an object as a function of time (t). The x-axis represents the time, the y-axis represents the acceleration of an object at each moment. In the example shown, an object has no acceleration between t=0.0 s and 2.0 s as well as between t=4.0 s and 6.0 s. Between t=2.0 s and 4.0 s, the acceleration is +2.0 m/s², meaning the object is increasing its velocity. Between t=6.0 s and 10. s, the acceleration is - 1m/s², meaning the object is decreasing its velocity. a) Between t=2.0 s and t=4.0 s, how much did the velocity change? b) Between t=6.0 s and t=10.0 s, how much did the velocity change? c) The change in velocity since t=0.0 s at each moment is equal to the area under the a-t curve from t=0.0 s to the time at which we want to know the velocity. Does this match with the Eq1?
d) (optional) Assuming v(t=0.0)=0.0 m/s, draw a v-t (velocity-time) graph with time on the x- axis and v on the y-axis. Don't forget the units! A09 [Qual] The figure shows the velocity of a particle at any given moment in time. a) In what direction is the particle travelling at t=3.0 s? b) In what direction is the particle travelling at t=6.0 s? c) At what time does the particle reverse its direction? d) When is the acceleration the highest? e) When is the acceleration the lowest? A10 The figure shows the velocity of a particle at any given moment in time. a) When is the object moving forward? b) When is it moving backward? c) When is it stationary? d) When is it speeding up? e) When is it slowing down? f) When does it have positive acceleration? g) When does it have negative acceleration? A11 [Challenge] For the figure in A10: a) Calculate the average acceleration from 0-10 s. b) Calculate the displacement and the average velocity from 0-10 s. c) Calculate the distance and the average speed from 0-10 s. A12 [Qual] [Challenge]
For each time given, state if instantaneous velocity and instantaneous acceleration are positive or negative. a) t=1.0 s b) t=5.0 s c) t= 7.0 s A13 [Qual] [ extra practice] [Easy] For the s-t graph shown, select the correct description A. v < 0.0 m/s, a = 0.0 m/s² B. v > 0.0 m/s, a > 0.0 m/s² C. v > 0.0 m/s,, a = 0.0 m/s² A14 [Qual] [ extra practice] [Easy] For the s-t graph shown, select the correct description A. v < 0.0 m/s, a = 0.0 m/s² B. v > 0.0 m/s, a > 0.0 m/s² C. v > 0.0 m/s,, a = 0.0 m/s² A15 [Qual] [Easy] [ extra practice] For each of the curves shown, state whether the object is speeding up, slowing down or going at constant velocity.
x A C B E t D F B Formulas with constant acceleration and freefall B01 [Qual] [Easy] A chocolate cake is thrown straight up. At the top of its trajectory its velocity is zero. a) Is its acceleration (at the top of its trajectory) positive, negative, or zero? b) As the object comes back down, is its acceleration positive, negative, or zero? B02 [Qual] [Easy] A superball is dropped to the floor and bounces back up. For an instant while it is in contact with the floor it momentarily stopped. Is its acceleration at that moment positive, negative, or zero? B03 [Easy] A ball is thrown vertically from ground level to rise to a maximum height of 50.0 m. a) What speed does the ball have when it was thrown? b) How long will it be in the air? c) Sketch a s-t, v-t and a-t graph of the ball. On the first two graphs, indicate the time at which 50 m is reached. B04 [Easy] A bus slows with constant acceleration from 25.0 m/s to 15.0 m/s and moves 50.0 m in the process. a) How much further does it travel before coming to a stop? b) How long does it take to stop from 24.0 m/s?
B05 A bolt is dropped from a bridge under construction, falling 90.0 m to the valley below the bridge. a) What is its speed when it begins the last 20.0m of its fall? b) What is its speed right before it lands on the valley beneath the bridge? c) In how much time does it pass through the last 20.0 m of its fall? B06 A ball thrown down from a balcony lands in 0.800 s at a speed of 13.0 m/s. Find: a) the initial velocity b) the height from which it was thrown c) the time to land if it were thrown up from the balcony with the same initial speed. B07 A car travelling 54.0 km/h towards west is 24.0 m from a wall when the driver slams on the brakes. a) What must its acceleration be in order for it to stop just in front of the wall? b) What is the direction of the acceleration? B08 A bus starts from rest at the origin and accelerates at 2.00 m/s 2 for 3.00 s. It moves at constant velocity for 2.00 s and then has an acceleration of -3.00 m/s 2 for 2.00 s. Plot the v-t and the x-t graph. Take x = 0.00 m and v = 0.00 m/s at t = 0.00 s. B09 [Challenge] A car moves along a horizontal road through a distance of 900. m. Through the first 1/4 of that distance, its acceleration is +6.25 m/s 2. Through the next 1/4 of that distance, it moves with constant velocity. Through the last half of that distance, its acceleration is -2.08 m/s 2. a) What is its travel time through the 900m? b) What is its maximum speed? c) What is its average speed through the 900m? d) What is its average acceleration through the 900m?
C. Catchup Problems (optional) C01 [Easy] Two stones are dropped from the same height, 1.0 s apart. a) Draw the x-t graph of the problem b) How long after the first stone begins to fall will the two stones be 10. m apart? C02 A police car is traveling along a straight road with 72 km/h when it is passed by a car. The car is 54 km/h faster than the police car. 5.0 seconds after it is passed, the police car starts to accelerate with a constant acceleration of 4.0 m/s². The passing car continues with constant velocity. a) Draw a v-t graph of the police car (continued line) and the passing car (dotted line) b) How far did the passing car travel in 5 seconds? c) How far ahead is the passing car after the 5 seconds? d) How many seconds after starting to accelerate does the police car catch up with the car? C03 A red ball is thrown vertically up at 5.0 m/s from a rooftop of height 100. m. 2.00 seconds later, a green ball is thrown down at 20. m/s from the same rooftop. a) Where and when do they meet? b) What are their velocities when they meet? C04 Two subway trains move toward each other on adjacent and parallel rail tracks. The distance between two train stations is 2200 m. The first subway train starts from rest from station A and accelerates at +1.0 m/s2 from rest. 30. s later, another subway train accelerates at 1.0 m/s2 from station B. Both trains have a top speed of 72 km/h. a) When and where do they meet? b) Draw one x-t graph for both trains.
C05 [Challenge] When a high-speed passenger train traveling at 180. km/h rounds a bend, the driver is shocked to see that a locomotive moving in the same direction has improperly entered onto the track from a siding and is a distance D = 676 m ahead. The locomotive is moving at 5.40 km/h. The driver of the high-speed train immediately applies the brakes. a) If the emergency deceleration of the train has a magnitude of 1.00 m/s 2, what would be the stopping distance of the train? b) What must be the magnitude of the resulting constant deceleration if a collision is to be just avoided? C06 At t=0.00 s, one toy train is set rolling on a straight track with an initial position 15.0 cm from the right end of the track. It has an initial velocity of 3.50 cm/s to the left and constant acceleration of 2.40 cm/s 2 to the left. At the same moment, another train is rolling on an adjacent track. The second train has an initial position 10 cm from the left end of the track and a constant velocity of 5.50 cm/s to the right. Both tracks have the same length of 2.00 m. I-----------------------------------------<-TTTTT -----------------------------I I---------TTTTT->--------------------------------------------------------------I a) At what time, do the two trains have equal speeds? b) What are their speeds at that time in a)? c) At what time(s) do the trains meet each other? d) What are their locations at that time in c)? e) Explain the difference between question a and c as clearly as possible. C07 The graph shows the velocity of two cars in a race. Both cars start at the same position. a) Which car has the higher acceleration? b) Which car has the higher top speed? c) At what time does car B pass car A? d) What are your answers for above questions, if they don t start from the same position?
C08 [Challenge] The Metro-Stations Atwater and Guy-Concordia are 800. m apart. A traditional MR-63 Metro train has the following specifications: v max =72.0 km/h, acceleration a acc =1.00 m/s², deceleration a dec =1.00 m/s². a) Starting from v 0 =0.00 km/h, after how many seconds does the MR-63 train reach its maximum speed? b) How far did the train travel during that time? c) How long does it take to travel from the stop at Atwater to the stop Guy-Concordia with the MR-63? (Remember that the train stops at both stations!) For the new Metro trains, STM had the option to change from the rubber-tire system to a steel wheel on steel rail system. This would have allowed a higher maximum speed but lower acceleration and deceleration. d) How long would it take to travel from the stop at Atwater to the stop Guy-Concordia using a steel wheel on steel rail system with v max =144 km/h, a acc =0.50 m/s², a dec =0.50 m/s²? D. Basic 2 Dimensional Problems D01 [Qual] [Easy] A person s speed can stay the same as he or she rounds a corner and changes direction. Given this information, is the velocity constant for the person? Explain [1]. D02 [Qual] [Easy] [extra] A student writes, A bird that is diving for prey has a speed of 10m/s. [1] a) What is wrong with the student s statement? What has the student actually described? Explain. b) What is the speed of the bird? D03 [Qual] [Easy] Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain. D04 [Easy] An ion s position is initially at (5.00, 2.00, -3.00), and 10.0 s later it is at (-5.00, 2.00, 7.00), all in meters. a) In unit vector notation, what is its average velocity during the 10.0 s? b) What is it average speed during the 10.0s?
D05 A small airplane flies at a constant altitude with a constant speed of 60.0 km/h. First, it goes east for 20.0 minutes. Then it goes 30.0 degrees north of east for 30.0 minutes. a) the average velocity of the plane for these 50.0 minutes. b) the average acceleration of the plane for these 50.0 minutes c) the average speed of palne for these 50. 0minutes. D06 A certain rectangular playing field is 150 m long and 100 m wide, with its long sides lying North-south. Starting at the southeast corner, you walk all around it clockwise. It takes you 8 minutes to walk halfway around (ie to the opposite corner) and another 12 minutes to get back to where you started. Find: a) the average velocity during the first 8 min b) the average speed during the first 8 min c) the average velocity during the last 12 min d) the average speed during the last 12 min e) the average velocity during the whole trip f) the average speed during the whole trip D07 [Qual] For each of (a, b, c) below, answer Yes or No, AND: If yes, when will it happen? If no, why not? a) Can the average speed ever be equal to the magnitude of average velocity? b) Can the average speed ever be larger than the magnitude of average velocity? c) Can the average speed ever be smaller than the magnitude of average velocity? D08 [Qual] For each of the following, state YES (if it is possible) or NO (never possible) AND, If yes, under what conditions and give an example. If no, why not? a) An object moving eastwards but accelerating westward. b) An object with negative acceleration while gaining speed. c) An object with negative velocity and zero acceleration. d) An object with zero velocity but accelerating. E. Projectile Motion and relative motion in 2D E01 [Easy] An object is fired from the edge of a cliff 50.0 m high. Its initial velocity is 40.0 m/s in direction 36.9 degrees above the horizontal. a) Where and when does it reach its highest point? b) Find its landing speed and direction
E02 An athlete can jump 8.5 m far. Assume his waistline returns to the initial level. If the takeoff is at 30 to the horizontal, what is the initial speed? E03 [Qual] Two footballs are kicked from ground level. They have reached the same maximum height but football A landed further than football B. Ignoring the effects of air. a) Which football is in air for longer time? b) Which football has a greater initial vertical velocity? c) Which football has a greater initial horizontal velocity component? d) Which football has a greater initial speed? E04 [extra practice] A cannonball is shot from ground at an angle α. It leaves the cannon with a speed of 40 m/s and hits a target (on the ground) at distance d after 4 seconds. E05 E06 a) Draw a v-t graph for the cannonball in x (horizontal) and y (vertical) direction b) At what angle α was the cannon shot? c) At what horizontal distance d is the target from the cannon? d) What is the maximal altitude h the cannonball is reaching? A helicopter travels with a constant velocity of 36 km/h at an altitude of 45. m. A package is ejected horizontally from the back of the helicopter (backwards) with a speed of 18 m/s. Calculate the angle, as measured from the horizontal) at which the package hits the ground. An airplane has a speed of 80.0m/s and is diving a angle of 30.0 below the horizontal when the pilot releases a package. The horizontal distance between the release point and the point where the decoy strikes the ground is d= 700. m. a) How long is the package in the air? b) How high was the release point?
E07 A canon is fired at a castle at a distance of 2.0 km. The cannonball leaves the cannon with 200m/s. a) In order to hit the castle, at what angle above the horizontal does the cannon have to be fired? (Ignore all effects of the air) b) How long is the cannonball in the air before hitting the castle? E08 E09 E10 E11 A ball rolls off a 1.00-m-high table and lands 1.60 m along the floor. Find a) its initial speed. b) the time of flight. [extra practice] A basketball player attempts to make a basket from a distance of 8.00m. The basket is 3.00m high and the ball leaves her hands at a height of 1.80m. If she throws the ball at an angle of 40 degrees, what does the initial velocity of the ball need to be? A ball is thrown from ground level. Three seconds later it is moving horizontally at 15 m/s. Find the horizontal range and the angle of impact. You throw a set of keys from a window on the 4th floor to your friend. He catches it 1.50 s later at height 1.2m above ground, at a distance d = 4.25 m from the building, and at angle 80.0 with the horizontal. a) Find the speed of the keys just before your friend catches them. b) Find the height where the keys are thrown. c) What are the magnitude and angle relative to the horizontal of the velocity at which the ball is thrown? d) Is the angle above or below the horizontal? E12 [extra practice] A motorcycle goes out of control and slides off a steep embankment of height h=50.0m at angle 30 to the horizontal. It lands in a ditch at a horizontal distance r=25.0 m from the road. Find the initial speed of the motorcycle.
v o θ h E13 [Qual] A hunter fires a gun with a harmless sedative bullet at a monkey hanging from a tree. The hunter aims directly at the monkey and fires. But just as he fires, the monkey lets go of the branch and begins to fall, thinking that he will thus fall below the trajectory of the bullet. Does the monkey avoid the bullet? E14 A football is kicked off with a velocity of magnitude 10.0 m/s, at an angle of 50.0 to the horizontal. The launch point is at the base of an inclined surface with 25 degree from the horizontal
50 o 25 o a) Where does the football land on the incline? b) When it lands, what are the magnitude and angle of its displacement from the launch point? c) When it lands, what are the magnitude and angle of its velocity? E15 [Challenge] E16 The range equation for projectile motion is: x = v i 2 sin (2θ i ) g Useful trigonometric identities: sin(2θ) = 2 sin(θ) cos(θ), sin 2 (θ) + cos 2 (θ) = 1. a) Show (do not just write an answer, you need to justify it) at what angle(s) above the horizontal an object should be launched in order to have the highest range for any given speed. b) Develop the above range equation, based on the formulas for constant acceleration. ] [Qual] If someone is riding in the back of a pickup truck and throws a softball straight backward, is it possible for the ball to fall straight down as viewed by a person standing at the side of the road? Under what condition would this occur? How would the motion of the ball appear to the person who threw it? [1] E17 A seagull flies at a velocity of 9.00 m/s straight into the wind. (a) If it takes the bird 20.0 min to travel 6.00 km relative to the Earth, what is the velocity of the wind? (b) If the bird turns around and flies with the wind, how long will he take to return 6.00 km? (c) Discuss how the wind affects the total round-trip time compared to what it would be with no wind. [1] E18 A ship sets sail from Rotterdam, The Netherlands, heading due north at 7.00 m/s relative to the water. The local ocean current is 1.50 m/s in a direction 40.0º north of east. What is the velocity of the ship relative to the Earth? [1] F. Kinematics of uniform circular motion
F01 [Qual][Easy] Is it possible for speed to be constant while acceleration is not zero? Give an example of such a situation. [1] F02 [Easy] A bike wheel has a radius of 40 cm. How many rotations will the wheel make when the bike travels 10.0 km? F03 [Easy] The earth travels around the sun in one year. Assuming that the earth travels on a circular orbit with radius 1.496 10 11 m, calculate a) The average speed of the earth over one year. b) The average velocity of the earth over one year. F04 [Easy] [extra practice] What is the speed of a person standing at the equator relative to the center of the Earth? The equatorial radius of the Earth is 6,378,137 m. F05 In a conical pendulum, a bob is suspended at the end of a string and moves around a horizontal circle at a constant speed of 1.21 m/s. If the length of the string is 1.2 m and it makes an angle of 20 with the vertical, find the acceleration of the bob. F06 A car goes counter clockwise (seen from above) around a circular track of radius 0.5 km at a constant speed of 100 km/hr. a) What is the period of the motion (i.e. how much time to do one whole circle?) b) What is the velocity when the car is exactly north of the center? c) What is the average acceleration of the car from the instant it is NE of center until the instant it is NW of center? [Hint: a avg = Δv/Δt ] Give the magnitude and direction of this average acceleration. G. Advanced Calculus based (optional) G01 [Easy] The position of a particle is given by x = 5 + 7t - 2t 2 m. a) Evaluate x at 1s intervals between 0 and 5 s and then plot x versus t. b) What is the average velocity between 1 and 2 s? c) Find the instantaneous velocity at 3 s.
G02 [Easy] Given the following equation of the position of an object as a function of time s(t)=s i +v i t+½ a t 2 Show that the velocity of the object as a function of time is v(t)=v i +at. G03 [Easy] An object is moving in uniform circular motion. Its position changes according to the following equations: x = r cos(ωt) y = r sin(ωt) a) Show that the speed is v= r. b) Show that the centripetal acceleration is a= 2 r. Ref [1] OpenStax College, College Physics. OpenStax CNX. September 29, 2015 http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4.