Vertical Projectile Motion in One Dimension (1D)

Umzilikazi Secondary School GRADE 12 Vertical Projectile Motion in One Dimension (1D) REVISION DOCUMENT 2021 COMPILED BY: B. DLAMINI Umzilikazi Secondary School 2021

Vertical Projectile Motion in One Dimension (1D) • Explain what is meant by a projectile, i.e. an object upon which the only force acting is the force of gravity. • Use equations of motion to determine the position, velocity and displacement of a projectile at any given time. • Sketch position versus time (x vs. t), velocity versus time (v vs. t) and acceleration versus time (a vs. t) graphs for: o A free-falling object o An object thrown vertically upwards o An object thrown vertically downwards o Bouncing objects (restricted to balls) • For a given x vs. t, v vs. t or a vs. t graph, determine: o Position o Displacement o Velocity or acceleration at any time t • For a given x vs. t, v vs. t or a vs. t graph, describe the motion of the object: o Bouncing o Thrown vertically upwards o Thrown vertically downward Page 2 Umzilikazi Secondary School 2021

QUESTION 1 A stationary rocket on the ground is launched vertically upwards. After 4 s, the rocket's fuel is used up and it is 225,6 m above the ground. At this instant the velocity of the rocket is 112,8 m∙s-1. The diagram below shows the path followed by the rocket. Ignore the effects of air friction. Assume that g does not change during the entire motion of the rocket. Q v = 112,8 m∙s-1 P 225,6 m ground 1.1 Write down the direction of the acceleration of the rocket at point: 1.1.1 P (1) 1.1.2 Q (1) 1.2 At which point (P or Q) is the rocket in free fall? Give a reason for the answer. (2) 1.3 TAKING UPWARD MOTION AS POSITIVE, USE EQUATIONS OF MOTION to calculate the time taken from the moment the rocket is launched until it strikes the ground. (6) 1.4 Sketch a velocity versus time graph for the motion of the rocket from the moment it runs out of fuel until it strikes the ground. Take the time when the rocket runs out of fuel as t = 0 s. Indicate the following values on the graph: Velocity of the rocket when it runs out of fuel Time at which the rocket strikes the ground (5) [15] Page 3 Umzilikazi Secondary School 2021

QUESTION 2 An object is released from rest from a point X, above the ground as shown in the diagram below. It travels the last 30 m (BC) in 1,5 s before hitting the ground. Ignore the effects of air friction. ●X ●B 30 m ● Ground C 2.1 Name the type of motion described above. (1) 2.2 Calculate the: 2.2.1 Magnitude of the velocity of the object at point B (4) 2.2.2 Height of point X above the ground (5) After hitting the ground, the object bounces once and then comes to rest on the ground. 2.3 Sketch an acceleration-time graph for the entire motion of the object. (3) [13] Page 4 Umzilikazi Secondary School 2021

QUESTION 3 (Mar. 2012) A stone is thrown vertically upward at a velocity of 10 m·s-1 from the top of a tower of height 50 m. After some time the stone passes the edge of the tower and strikes the ground below the tower. Ignore the effects of friction. vi = 10 m·s-1 1,5 m 50 m y1 3.1 Draw a labelled free-body diagram showing the force(s) acting on the stone during its motion. (1) 3.2 Calculate the: 3.2.1 Time taken by the stone to reach its maximum height above the ground (4) 3.2.2 Maximum height that the stone reaches above the ground (4) 3.3 USING THE GROUND AS REFERENCE (zero position), sketch a position- time graph for the entire motion of the stone. (3) 3.4 On its way down, the stone takes 0,1 s to pass a window of length 1,5 m, as shown in the diagram above. Calculate the distance (y1) from the top of the window to the ground. (7) [19] Page 5 Umzilikazi Secondary School 2021

QUESTION 4 (Nov. 2018) In a competition, participants must attempt to throw a ball vertically upwards past point T, marked on a tall vertical pole. Point T is 3,7 m above the ground. Point T may, or may not, be the highest point during the motion of the ball. One participant throws the ball vertically upwards at a velocity of 7,5 ms-1 from a point that is 1,6 m above the ground, as shown in the diagram below. Ignore the effects of air resistance. T 7,5 ms-1 3,7 m 1,6 m ground 4.1 In which direction is the net force acting on the ball while it moves towards point T? Choose from: UPWARDS or DOWNWARDS. Give a reason for the answer. (2) 4.2 Calculate the time taken by the ball to reach its highest point. (3) 4.3 Determine, by means of a calculation, whether the ball will pass point T or not. (6) 4.4 Draw a velocity-time graph for the motion of the ball from the instant it is thrown upwards until it reaches its highest point. Indicate the following on the graph: The initial velocity and final velocity Time taken to reach the highest point (2) [13] Page 6 Umzilikazi Secondary School 2021

QUESTION 5 (DBE 2015) A hot air balloon is rising vertically at a constant velocity. When the hot air balloon reaches point A a few metres above the ground, a man in the hot air balloon drops a ball which hits the ground and bounces. Ignore the effects of friction. A GROUND The velocity-time graph below represents the motion of the ball from the instant it is dropped until after it bounces for the first time. The time interval between bounces is ignored. THE UPWARD DIRECTION IS TAKEN AS POSITIVE. USE INFORMATION FROM THE GRAPH TO ANSWER THE QUESTIONS THAT FOLLOW. 5.1 Write down the magnitude of the velocity of the hot air balloon. (1) 5.2 Calculate the height above the ground from which the ball was dropped. (3) Calculate the: 5.3 Time at the point P indicated on the graph (2) 5.4 Maximum height the ball reaches after the first bounce (3) 5.5 Distance between the ball and hot air balloon when the ball is at its maximum height after the first bounce (4) Umzilikazi Secondary School 2021

QUESTION 6 (Mar. 2017) A hot-air balloon moves vertically downwards at a constant velocity of 1,2 m∙s-1. When it reaches a height of 22 m from the ground, a ball is dropped from the balloon. Refer to the diagram below. 1,2 m·s-1 22 m ground Assume that the dropping of the ball has no effect on the speed of the hot-air balloon. Ignore air friction for the motion of the ball. 6.1 Explain the term projectile motion. (2) 6.2 Is the hot-air balloon in free fall? Give a reason for the answer. (2) 6.3 Calculate the time it takes for the ball to hit the ground after it is dropped. (4) When the ball lands on the ground, it is in contact with the ground for 0,3 s and then it bounces vertically upwards with a speed of 15 m∙s-1. 6.4 Calculate how high the balloon is from the ground when the ball reaches its maximum height after the first bounce. (6) [14] Page 8 Umzilikazi Secondary School 2021

QUESTION 7 (Mar. 2013) A ball of mass 0,2 kg is dropped from a height of 0,8 m onto a hard floor. It bounces to a maximum height of 0,6 m. The floor exerts a force of 50 N on the ball. Ignore the effects of friction. 7.1 Write down the magnitude and direction of the force that the ball exerts on the floor. (2) 7.2 Calculate the: 7.2.1 Velocity at which the ball strikes the floor (4) 7.2.2 Time that the ball is in contact with the floor if it bounces off the floor at a speed of 3,43 m·s-1 (4) 7.3 The ball takes 0,404 s from the moment it is dropped until it strikes the floor. Sketch a graph (not to scale) of position versus time representing the entire motion of the ball. USE THE GROUND AS ZERO REFERENCE. Indicate the following on the graph: Height from which the ball is dropped Height reached by the ball after the bounce Time at which the ball bounces off the floor (5) [15 ] Page 9 Umzilikazi Secondary School 2021

QUESTION 8 (Nov. 2013) A ball of mass 0,15 kg is thrown vertically downwards from the top of a building to a concrete floor below. The ball bounces off the floor. The velocity versus time graph below shows the motion of the ball. Ignore the effects of air friction. TAKE DOWNWARD MOTION AS POSITIVE. 8.1 From the graph, write down the magnitude of the velocity at which the ball bounces off the floor. (1) 8.2 Is the collision of the ball with the floor ELASTIC or INELASTIC? Refer to the data on the graph to explain the answer. (3) 8.3 Calculate the: 8.3.1 Height from which the ball is thrown (4) 8.3.2 Magnitude of the impulse imparted by the floor on the ball (3) 8.3.3 Magnitude of the displacement of the ball from the moment it is thrown until time t (4) 8.4 Sketch a position versus time graph for the motion of the ball from the moment it is thrown until it reaches its maximum height after the bounce. USE THE FLOOR AS THE ZERO POSITION. Indicate the following on the graph: The height from which the ball is thrown Time t (4) [19] Page 10 Umzilikazi Secondary School 2021

QUESTION 10 (DBE 2017) A ball is projected vertically upwards with a speed of 10 m∙s-1 from point A, which is at the top edge of a building. The ball hits the ground after 3 s. It is in contact with the ground for 0,2 s and then bounces vertically upwards, reaching a maximum height of 8 m at point B. See the diagram below. Ignore the effects of friction. 10 m∙s-1 A B 8m ground 10.1 Why is the ball considered to be in free fall during its motion? (2) 10.2 Calculate the: 10.2.1 Height of the building (3) 10.2.2 Speed with which the ball hits the ground (3) 10.2.3 Speed with which the ball leaves the ground (3) 10.3 Draw a velocity versus time graph for the complete motion of the ball from A to B. Show the following on the graph: The magnitude of the velocity with which it hits the ground The magnitude of the velocity with which it leaves the ground The time taken to reach the ground, as well as the time at which it leaves the ground (4) [15] Page 11 Umzilikazi Secondary School 2021

QUESTION 11 (Mar. 2018) A ball is thrown vertically downwards from the top of a building and bounces a few times as it hits the ground. The velocity-time graph below describes the motion of the ball from the time it is thrown, up to a certain time T. Take downwards as the positive direction and the ground as zero reference. The graph is NOT drawn to scale. The effects of air friction are ignored. 11.1 Write down the speed with which the ball is thrown downwards. (1) 11.2 ALL parts of the graph have the same gradient. Give a reason for this. (2) 11.3 Calculate the: 11.3.1 Height from which the ball is thrown (3) 11.3.2 Time (T) shown on the graph (4) 11.4 Write down the: 11.4.1 Time that the ball is in contact with the ground at the first bounce (1) 11.4.2 Time at which the ball reaches its maximum height after the first bounce (2) 11.4.3 Value of X (1) 11.5 Is the collision of the ball with the ground elastic or inelastic? Give a reason for the answer using information in the graph. (2) [16] Page 12 Umzilikazi Secondary School 2021

QUESTION 12 (DBE 2018) In the diagram below, point A is at the top of a building. Point B is exactly halfway between the point A and the ground. Ignore air resistance. ●A ●B ground 12.1 Define the term free fall. (2) A ball of mass 0,4 kg is dropped from point A. It passes point B after 1 s. 12.2 Calculate the height of point A above the ground. (3) When the ball strikes the ground it is in contact with the ground for 0,2 s and then bounces vertically upwards, reaching a maximum height at point B. 12.3 Calculate the magnitude of the velocity of the ball when it strikes the ground. (3) 12.4 Calculate the magnitude of the average net force exerted on the ball while it is in contact with the ground. (6) [14] Page 13 Umzilikazi Secondary School 2021

QUESTION 13 (Nov. 2014) A ball, A, is thrown vertically upward from a height, h, with a speed of 15 m∙s-1. AT THE SAME INSTANT, a second identical ball, B, is dropped from the same height as ball A as shown in the diagram below. Both balls undergo free fall and eventually hit the ground. 15 m∙s-1 B A h Ground 13.1 Explain the term free fall. (2) 13.2 Calculate the time it takes for ball A to return to its starting point. (4) 13.3 Calculate the distance between ball A and ball B when ball A is at its maximum height. (7) 13.4 Sketch a velocity-time graph in the ANSWER BOOK for the motion of ball A from the time it is projected until it hits the ground. Clearly show the following on your graph: The initial velocity The time it takes to reach its maximum height The time it takes to return to its starting point (4) [17] Page 14 Umzilikazi Secondary School 2021

QUESTION 14 (Nov. 2015) Ball A is projected vertically upwards at a velocity of 16 m∙s-1 from the ground. Ignore the effects of air resistance. Use the ground as zero reference. 14.1 Calculate the time taken by ball A to return to the ground. (4) 14.2 Sketch a velocity-time graph for ball A. Show the following on the graph: (a) Initial velocity of ball A (b) Time taken to reach the highest point of the motion (c) Time taken to return to the ground (3) ONE SECOND after ball A is projected upwards, a second ball, B, is thrown vertically downwards at a velocity of 9 m∙s-1 from a balcony 30 m above the ground. Refer to the diagram below. B 9 m∙s-1 30 m 16 m∙s-1 A ground 14.3 Calculate how high above the ground ball A will be at the instant the two balls pass each other. (6) [13] Page 15 Umzilikazi Secondary School 2021

QUESTION 15 (Mar. 2016) A man throws ball A downwards with a speed of 2 m∙s-1 from the edge of a window, 45 m above a dam of water. One second later he throws a second ball, ball B, downwards and observes that both balls strike the surface of the water in the dam at the same time. Ignore air friction. 15.1 Calculate the: 15.1.1 Speed with which ball A hits the surface of the water (3) 15.1.2 Time it takes for ball B to hit the surface of the water (3) 15.1.3 Initial velocity of ball B (5) 15.2 On the same set of axes, sketch a velocity versus time graph for the motion of balls A and B. Clearly indicate the following on your graph: Initial velocities of both balls A and B The time of release of ball B The time taken by both balls to hit the surface of the water (5) [16] Page 16 Umzilikazi Secondary School 2021

QUESTION 15 (DBE 2016) Ball A is projected vertically upwards from the ground, near a tall building, with a speed of 30 m∙s-1. Ignore the effects of air friction. 15.1 Explain what is meant by a projectile. (2) 15.2 Calculate: 15.2.1 The total time that ball A will be in the air (4) 15.2.2 The distance travelled by ball A during the last second of its fall (4) 15.3 TWO SECONDS after ball A is projected upwards, ball B is projected vertically upwards from the roof of the same building. The roof the building is 50 m above the ground. Both balls A and B reach the ground at the same time. Refer to the diagram below. Ignore the effects of air friction. B 50 m 30 m∙s-1 A ground Calculate the speed with which ball B was projected upwards from the roof. (4) 15.4 Sketch velocity-time graphs for the motion of both balls A and B on the same set of axes. Clearly label the graphs for balls A and B respectively. Indicate the following on the graphs: (a) Time taken by both balls A and B to reach the ground (b) Time taken by ball A to reach its maximum height (4) [18] Page 17 Umzilikazi Secondary School 2021

QUESTION 16 (Nov. 2017) Stone A is projected vertically upwards at a speed of 12 m∙s-1 from a height h above the ground. Ignore the effects of air resistance. 16.1 Calculate the time taken for stone A to reach its maximum height. (3) At the same instant that stone A is projected upwards, stone B is thrown vertically downwards from the same height at an unknown speed, v. Refer to the diagram below. 12 m∙s-1 A B v h ground When stone A reaches its maximum height, the speed of stone B is 3v. 16.2 Calculate the speed, v, with which stone B is thrown downwards. (4) At the instant stone A passes its initial position on its way down, stone B hits the ground. 16.3 Calculate the height h. (3) 16.4 Sketch velocity-time graphs for the complete motions of stones A and B on the same set of axes. Label your graphs for stones A and B clearly. Show the following on the graphs: The time taken for stone A to reach its maximum height The velocity with which stone B is thrown downwards (4) [14] Page 18 Umzilikazi Secondary School 2021

QUESTION 17 (Nov. 2019) Stone A is thrown vertically upwards with a speed of 10 m∙s-1 from the edge of the roof of a 40 m high building, as shown in the diagram below. Ignore the effects of air friction. Take the ground as reference. 10 m∙s-1 A 40 m ground 17.1 Define the term free fall. (2) 17.2 Calculate the maximum HEIGHT ABOVE THE GROUND reached by stone A. (4) 17.3 Write down the magnitude and direction of the acceleration of stone A at this maximum height. (2) Stone B is dropped from rest from the edge of the roof, x seconds after stone A was thrown upwards. 17.4 Stone A passes stone B when the two stones are 29,74 m above the ground. Calculate the value of x. (6) Page 19 Umzilikazi Secondary School 2021

17.5 The graphs of position versus time for part of the motion of both stones are shown below. Which of labels a to h on the graphs above represents EACH of the following? 17.5.1 The time at which stone A has a positive velocity (1) 17.5.2 The maximum height reached by stone A (1) 17.5.3 The time when stone B was dropped (1) 17.5.4 The height at which the stones pass each other (1) [18] Page 20 Umzilikazi Secondary School 2021