Chapter (3) Motion in One Dimension Pro. Mohammad Abu Abdeen Dr. Galal Ramzy
Chapter (3) Motion in one Dimension We begin our study o mechanics by studying the motion o an object (which is assumed to be ery small in dimensions compared to any other dimension in the problem under consideration and we may reer to it by particle) in one direction. i.e., the particle moes along a straight line which will be taken to be the x - axis unless stated otherwise. Displacement, Velocity and Acceleration Let s consider a car that is moing along the positie x-direction as shown in Figure. The point O is called the origin. i.e., it is the reerence point rom which all distances are measured. The car is assumed to start its motion rom the origin at which we start our clock. It is moing toward the right, and is located at distance x rom the origin at time t then the car passes through another point which is at a distance x rom the origin ater time t. Now we may deine the ollowing: Displacement It is the distance moed in a speciied direction. displacement x x x, The symbol (pronounced as delta) represents the change in x. Note that the displacement may positie or negatie depending on the direction o motion. A negatie displacement means that the object is moing in the negatie x - direction and ice ersa. Distance It is the magnitude o displacement distance = x x x Pro. Mohammad Abu Abdeen Dr. Galal Ramzy
and it must be positie. Howeer, to clariy the dierence between displacement and distance suppose we hae traelled rom city to another (a total distance o km) in a round trip. Our total displacement will be zero while our total distance traelled will be 44 km. Velocity is deined as the rate (rate means change with time) at which displacement changes. Like displacement, elocity may positie or negatie depending on the direction o motion. Speed is the magnitude o elocity Aerage Speed is deined as the ratio between the total distance traeled to the time taken. i.e., a Total distance moed x x x Total time taken t t t Instantaneous Speed it is the speed at a certain instant and is deined as inst x x x dx t t t dt lim lim t tt Simply it is the speed at ery short time interal (instant). I you are driing a car, it is the reading o the speedometer at the moment you look at it. It may change rom an instant to another depending on the press you made on the acceleration paddle. Uniorm elocity I the instantaneous speed is the same during a certain time interal then it is called a uniorm speed. (i.e. it is always constant or has the same alue) Example 3. The position o particle that moes in a straight line changes with time according to the relation x 3 5 t where x is measured in meters and t in seconds. (a) Find the distance traelled between t = s, and t = 3 s. (b) Find the aerage speed during the same time interal. (c) In which direction the particle is moing? (d) What is the instantaneous speed at t = 3 s? (a) At t = s, the particle will be at Pro. Mohammad Abu Abdeen 3 Dr. Galal Ramzy
3 x 5 m t s While at t = 3 s, the particle will be at 3 x 5 3 45 m 3 t s So the distance traelled by the particle in this time interal will be x x x 45 85 m (b) The aerage speed can be ound using the deinition a x x x 45 t t t 3 85 m/s (c) Since the displacement o the particle during this time interal was positie, the particle is moing in the positie x direction.. (d) The instantaneous elocity o the particle at any time could be ound using its deinition inst dx dt d dt t 3 5 45 t m/s At t = 3 s, the particle s speed will be 45 3 45 m/s inst Practice Exercise 3. The speed o light, c is 3 8 m/s. (a) How long does it take or light to trael rom the sun to the earth, a distance o.5 m? (b) How long does it take or light to trael rom the moon to the earth, a distance o 3.84 8 m? (c) A light-year is a unit o distance equal to that traelled by light in year. Conert light-year into kilometers. Example 3. A space crat is traelling directly toward the sun. At time t it is at x = 3 m relatie to the sun. Exactly one year later it is at x =. m. ind its displacement and aerage elocity. Pro. Mohammad Abu Abdeen 4 Dr. Galal Ramzy
I we choose t to be zero, then t = y = 3.6 7 s, then the displacement x x x. 3. 9 m, The negatie sign indicates that the crat is approaching the sun. Also, the aerage elocity is a x x x t t t. 3 7 3. 6 4. 85 m/s 8. 5 km/s Practice Exercise 3. A car traels in a straight line with an aerage elocity o 8 km/h or.5 h and then with an aerage elocity o 4 km/h or.5 h. (a) what is the total displacement or the 4-h trip? (b) What is the aerage elocity or the total trip? Example 3.3 In a 4 m ootrace, the runner coer the st m with an aerage elocity o m/s and the nd m with an aerage elocity o m/s. What s his aerage elocity or the entire 4 m? The total distance coered, x 4 m The total time taken, x x t t t 6. 7 = 36.7 s So the aerage elocity or the entire 6 m is a x 4. 9 m/s t 36. 7 Practice Exercise 3.3 A car traels 8 km in a straight line. I the st 4 km is coered with an aerage elocity 8 km/h, and the total trip takes. h, what was the aerage elocity during the nd 4 km? Pro. Mohammad Abu Abdeen 5 Dr. Galal Ramzy
Distance Time graphs Sometimes, it is ery helpul to draw a graph representing the relation between distance and time or the problem under consideration. Dierent types o such graphs exist, as will be shown. Figure (a) shows that the particle elocity does not change with time, rom which we can conclude that the particle is at rest and hence its instantaneous and aerage elocities are zero. Figure (b) shows that the particle s position is changing linearly with time. The instantaneous elocity is the slope o this line by the geometrical interpretation o the irst deriatie Figure (a) inst dx AB slope dt OB Figure (b) In this case, the object is moing with a uniorm (constant) elocity (since a straight line has one slope) Also, the aerage elocity is equal to the instantaneous speed and the motion is called uniorm motion. Note that i the straight line does not pass through the origin and intersects the distance axis, this only alters the starting point o the motion but the motion still uniorm and so does the elocity. Figure (c) is the general case o a distance time graph, in which the relation is a cure, and hence the instantaneous elocity changes rom one point to another along the cure, and the resulting motion is not uniorm in this case. To ind the instantaneous elocity at a certain instant, it is equal to the slope o the cure at that instant. Figure (c) Pro. Mohammad Abu Abdeen 6 Dr. Galal Ramzy
GE Velocity tan EF Acceleration It is the time rate o change o elocity, and just as we hae or the elocity, there are two types o acceleration, the aerage acceleration; a a t and the instantaneous acceleration; a ins d lim = t t dt I the acceleration is positie, the elocity increases. But i the acceleration is negatie the elocity decreases and is reerred to as deceleration or retardation. Also, the acceleration is said to be uniorm, i the time rate o change o elocity is constant. Velocity - Time graphs Consider a body which is moing with elocity 6 m/s, and ater 5 s it has moed a distance equal to 3 m. I we plot a graph o elocity against time or that body, it will be a straight line parallel to the x axis, in which the distance traeled by the body equals to the area under the cure. i.e., the area between the x - axis and the elocity time graph. N.B. The acceleration o the body equals zero since its speed does not change with time. Uniormly Accelerated Motion In this type o motion, the acceleration is constant, and is represented graphically by a straight line, whose slope represents the speed o the body. The point at which the line intersects the elocity axis represents the initial speed o the body. Pro. Mohammad Abu Abdeen 7 Dr. Galal Ramzy
Example 3.4 Calculate the total distance traeled by a body whose elocity - time graph is shown in Fig. analytically and graphically Solution [] Analytical Method Since the speed o the body is not uniorm we must calculate the aerage speed o the body irst a i 3 7 m/s thereore the total distance traeled will be [] Graphical Method x a t 7 4 8 m The area between the elocity time graph and the time axis equals to the area o the triangle ABM and the area o the rectangle OCMA that equals to the distance traeled. Then The distance traelled = 4 8 = 6 + = 8 m which is the same result obtained beore. Practice Exercise 3.4 (a) Draw the elocity- time graph or a body that starts its motion along a straight line with a speed 4 m/s, and keep moing with acceleration.5 m/s or 6 s. (b)use the graph to ind the aerage speed and the total distance traelled..example 3.5 A cheetah can accelerate rom to 96 km/h in s, whereas a Corette requires 4.5 s. Compute the aerage acceleration or the cheetah and Corette and compare them with Pro. Mohammad Abu Abdeen 8 Dr. Galal Ramzy
the ree all acceleration due to graity, g = 9.8 m/s. Solution First we hae to conert the unit o the elocity into the standard SI unit which is m/s, km m 96 96 6. 67 m/s h 66 s or cheetah a a 6.67 3. 3 m/s t and comparing this alue to the acceleration due to graity, Doing the same or the Corette a a 3. 3. 36 g 98. a a 6.67 t 4.5 5. 9 m/s and comparing this alue to the acceleration due to graity, a a 5. 9 g 98. 6. Practice Exercise 3.5 A car is traelling at 45 km/h at time t =. It accelerates at a constant rate o km/h (a) How ast is it traelling at t = s? (b)at what time is the car traelling at 7 km/h Example 3.6 The elocity o a particle moing along the x - axis aries in time according to the relation x 4 5 t m/s, where t is in seconds. (a) Find the aerage acceleration in the time interal between t = and t = s. (b) Determine the acceleration at t = s. Solution (a) First we hae to ind the initial and inal elocities o the particle Pro. Mohammad Abu Abdeen 9 Dr. Galal Ramzy
i 4 5 4 m/s t & By deinition o the aerage acceleration 4 5 m/s t a a i 4 = = m/s t t The negatie sign indicates that the particle is decelerating in this time interal. (b) The acceleration at a certain instant is ound rom the deinition o the instantaneous acceleration a d = = - m/s ins ts dt ts t Practice Exercise 3.6 A particle moes along the x - axis according to the equation; x t t 3, where x is in meters and t in seconds. At t = 3 s, ind (a) the position, (b) the elocity and (c) the acceleration. Equations o uniormly accelerated motion The equations o motion are mathematical relations that relate the parameters o motion to each other namely, the initial elocity, the inal elocity, the acceleration and time. In soling problems using these equations, we choose the most appropriate equation or calculating the required parameters, the equation with one unknown only, which is the required parameter. First Equation o motion Consider a body which is gien an initial elocity i and has an acceleration a, then ater time t, its inal speed ; is gien by a i t t Pro. Mohammad Abu Abdeen Dr. Galal Ramzy
Or a t i Second Equation o motion I a body is to hae an initial elocity i and inal elocity then its aerage elocity is a i and since a t, thereore i i a t a i a t And the distance traeled by the body ater time t is gien by x x i at i t a t Third Equation o Motion Since Then a t a t i i i and since or Then x at i t a t a t a t i i Pro. Mohammad Abu Abdeen Dr. Galal Ramzy
a x x i The aboe three equations together with the aerage speed equation are known as the equations o motion. Example 3.7 A space ehicle returning rom the moon enters normally aboe its landing position into the atmosphere with a speed o km/s, and is brought to rest with an acceleration not to exceed m/s. (a) Will the ehicle land saely using the aboe parameters? (b) I the answer o part (a) indicates that the ehicle will not land saely on the earth s surace, discuss how to sole this situation. Solution The time taken by the ehicle to come to rest can be calculated rom the deinition o the acceleration a t t i i t While the aerage elocity during the trip t s a 55. km/s = 55 m/s That is the distance required or the ehicle to come to a sae stop at the earth s surace is x at 55 65 m = 65 km Then the necessary distance is 65 km which is a problem since the thickness o the atmosphere is only 6 km which means that the crat cannot enter Pro. Mohammad Abu Abdeen Dr. Galal Ramzy
ertically and the path must be gently sloping making an angle with the horizon. To ind the alue o we notice rom the ig. That 6 sin. 9 57. 65 At an angle less than this, the ehicle would skip back into space, and at an angle greater than 7. o the acceleration orces would be too high and the crat will hae a crash with the ground. Practice Exercise 3.7 A jet lands on an aircrat carrier at nearly 63 m/s. (a) What is the acceleration (assumed constant) i it stops in s due to an arresting cables that snags the airplane and bring it to a stop? (b) I the plane touches at xi, what is the inal position o the plane? Example 3.8 A stone is thrown ertically upward with an initial elocity 4 m/s. Find: (a) Maximum height reached. (b) Time taken beore reaching the ground. Solution (a) When the stone reaches the maximum height it comes to rest momentary and thus its inal speed becomes zero. g x x g x o x. 4 m g 9 8 (b) And the time taken beore reaching this height is ound rom x x g t 4 9. 8 t Pro. Mohammad Abu Abdeen 3 Dr. Galal Ramzy
t 4 98.. 43 s & the time taken beore the body reaches the ground is Time t. 43. 86 s Practice Exercise 3.8 On a highway, a drier sees a stalled ehicle and brakes his car to stop with an acceleration 5 m/s. What is the car s stopping distance i its initial speed is (a)5 m/s or (b) 3 m/s. Example 3.9 A car starts rom rest is accelerated uniormly at the rate m/s or 6 s. It then maintains a constant speed or hal a minute. The brakes are then applied and the ehicle retarded to rest in 5 s. Find the maximum speed reached in km/h and the total distance traeled in meters. First Stage, = m/s a, t = 6 s, The inal speed reached is a t 6 m/s t6s and the distance traelled is x x x t a t xi x x a t 6 36 m. Second Stage m/s, a, t 3 s. x Then xii x x t a t 3 36 m Third Stage m/s, =, t = 5 s, Pro. Mohammad Abu Abdeen 4 Dr. Galal Ramzy
The acceleration during this stage is a 5 4 = t = -. m/s and using the third equation o motion and the total distance traeled is a x x x 4.. III xiii 3 m 4 xtot xi xii xiii 46 m while the maximum elocity reached by the object is m/s 43. km/h Practice Exercise 3.9 Find the total distance traelled by the car in example.9 graphically. Freely Falling Objects From our daily lie experience, i an object is let to all in the graitational ield o the earth, it will moe toward the earth which is always acting toward the center o the earth (e.g, always acting downward). The acceleration with which the object approaches the earth s surace is reerred to as the acceleration due to graity g and was ound to hae the alue o 9.8 m/s at the equator and it decreases slightly as we moe towards the poles due to the eect o rotation o the earth around its axis. Pro. Mohammad Abu Abdeen 5 Dr. Galal Ramzy
A reely alling object is deined as any object moing reely under the inluence o graity only. Any reely alling object experience acceleration equals to g regardless o the initial conditions o its motion is changed. An experiment was irst perormed by Galileo Galilei (564-64) and you can try it yoursel, bring two dierent objects, one is heay (e.g. a small rubber ball) and the other is light (e.g., a piece o paper). Hold both objects in your hands and let them all simultaneously as possible. You will notice that they take the same time to reach the earth s surace (i we neglect the resistance o air) and also they will hae the same speed as will be shown. Now we shall calculate the speed with which a reely alling object hit the earth s surace rom a height h aboe the earth s surace with zero initial elocity. Using the nd equation o motion, i we consider the direction o motion to be the x-direction; a x x x Using the initial conditions, a g, x x h o x i g h = g h This speed is called the ree all speed. Obiously, the ree all speed is independent o the mass o the reely alling object and it depends only on the height rom which the object alls and the alue o the acceleration due to graity at the place. Example 3. A one - pound coin is dropped rom Cairo tower. It starts rom rest and alling reely. Compute its position and elocity ater s, s, and 3 s. Pro. Mohammad Abu Abdeen 6 Dr. Galal Ramzy
Taking the ertical direction as the y - direction with the positie direction pointing upward with the origin at the dropping point, and remembering that the reely alling body always moe with the acceleration due to graity g, which always acting downward, thereore, ay g 98. m/s and y Applying the equations o motion y y t a t y y y t 9. 8t y 98. t & y y ay t y g t y 98. t At t = s, At t = s, At t = 3 s, 9 8 4 9 y.. m 9 8 9 6 9. 8 9. 8 m/s y.. m y 9. 8 9. 6 m/s y 9 8 44 y. 3. m 9. 8 3 9. 4 m/s The negatie sign appears since we hae taken the positie direction to be upward. Practice Exercise 3. A rocket moes straight upward, starting rom rest with an acceleration o 9.4 m/s. It runs out o uel at the end o 4 s and continues to coast upward, reaching a maximum height beore alling back to Earth. (a) Find the rocket s elocity and position at the y Pro. Mohammad Abu Abdeen 7 Dr. Galal Ramzy
end o 4 s. (b) Find the maximum height the rocket reaches. (c) Find the elocity the instant beore the rocket crashes on the ground. Problems on Chapter (3) P 3. A motorist traels rom town A to town B, 45 km distance in 3 h 45 min. Find the aerage speed in (a) m/s ; (b) km/h. P 3. A car traels at a uniorm speed o m/s, or 5 s. The brakes are then applied and the car comes to rest with uniorm retardation in a urther 8 s. How ar does the car trael ater the brakes are then applied? P 3.3 A motorist traelling at 9 km/h, applies the brakes and comes to rest with uniorm retardation in s. Calculate the retardation in m/s. P 3.4 An electric train moing at km/h accelerates to a speed o 3 km/h in s. Find the aerage acceleration in m/s and the distance traeled in meters during the period o acceleration. P 3.5 A small iron ball is dropped rom the top o a ertical cli and takes.5 s to reach the sandy beach below. Find: (a) The elocity with which it strikes the sand. (b) The height o the cli. (c) I the ball penetrates the sand to a depth o.5 cm, calculate the aerage retardation. P 3.6 A car is speeding at 5 m/s in a school zone. A police car starts rom rest just as the speeder passes and accelerates at a constant rate o 5 m/s. (a) When does the police car catch the speeding car? (b) How ast is the police car traelling when it catches up with the speeder? P 3.7 Draw a graph o elocity against time or a body which starts with an initial elocity 4 m/s, and continues to moe with an acceleration.5 m /s or 6 s. Show how you can ind rom the graph (a) The aerage speed. (b) The distance moed in 6 s. Pro. Mohammad Abu Abdeen 8 Dr. Galal Ramzy
P 3.8 A boy climbs a tree to get better iew at an outdoor graduation ceremony. Unortunately, he leaes his binoculars behind. Another boy throws them up to the st boy but his strength is greater than his accuracy. The binoculars pass the st boy outstretched hand ater.69 s and again.68 later. How high is the st boy? Pro. Mohammad Abu Abdeen 9 Dr. Galal Ramzy