PHYSICS IN ACTION
path of head
path of centre of mass (parabolic)
• Projectiles move in parabolic paths that can be analysed by considering the horizontal and vertical components of the motion.
• If air resistance is ignored, the only force acting on a projectile is its weight, i.e. the force of gravity Fg. This results in the projectile having a vertical acceleration
will fall at exactly the same rate, and in the same time, as an object falling vertically from the same height.
• At the point of maximum height, a projectile is moving horizontally. Its velocity at this point is given by the horizontal component of its velocity as the vertical component equals zero.
Figure 1.21 shows a food parcel being dropped from a plane moving at a constant velocity. If air resistance is ignored the parcel falls in a parabolic arc.
It would continue moving horizontally at the same rate as the plane; that is, as the parcel fell it would stay directly beneath the plane until the parcel hit the ground. The effect of air resistance is also shown. Air resistance (or drag) is a retarding force and it acts in a direction that is opposite to the motion of the projectile. If air resistance is taken into account, there are now two forces acting—weight Fgand air resistance Fa. Therefore the resultant force ΣF that acts on the projectile is not vertically down. The magnitude of the air resist-ance force is greater when the speed of the body is greater.
1.5 SUMMARY PROJECTILE MOTION
1.5 QUESTIONS
For the following questions you may assume that the
acceleration due to gravity is 9.80 m s−2and the effects of air resistance may be ignored unless otherwise stated.
1 A golfer practising on a range with an elevated tee 4.9 m above the fairway is able to strike a ball so that it leaves the club with a horizontal velocity of 20 m s−1.
a How long after the ball leaves the club will it land on the fairway?
b What horizontal distance will the ball travel before striking the fairway?
c What is the acceleration of the ball 0.5 s after being hit?
d Calculate the speed of the ball 0.80 s after it leaves the club.
e With what speed will the ball strike the ground?
2 A bowling ball of mass 7.5 kg travelling at 10 m s−1rolls off a horizontal table 1.0 m high.
a Calculate the ball’s horizontal velocity just as it strikes the floor.
b What is the vertical velocity of the ball as it strikes the floor?
c Calculate the velocity of the ball as it reaches the floor.
d What time interval has elapsed between the ball leaving the table and striking the floor?
e Calculate the horizontal distance travelled by the ball as it falls.
The following information applies to questions 3–8. A senior physics class conducting a research project on projectile motion constructs a device that can launch a cricket ball. The launching device is designed so that the ball can be launched at ground level with an initial velocity of 28 m s−1at an angle of 30° to the horizontal.
3 Calculate the horizontal component of the velocity of the ball:
a initially b after 1.0 s c after 2.0 s.
4 Calculate the vertical component of the velocity of the ball:
a initially b after 1.0 s c after 2.0 s.
5 a At what time will the ball reach its maximum height?
b What is the maximum height that is achieved by the ball?
c What is the acceleration of the ball at its maximum height?
6 a At which point in its flight will the ball experience its minimum speed?
b What is the minimum speed of the ball during its flight?
c At what time does this minimum speed occur?
7 a At what time after being launched will the ball return to the ground?
b What is the velocity of the ball as it strikes the ground?
c Calculate the horizontal range of the ball.
8 If the effects of air resistance were taken into account, which one of the following statements would be correct?
A The ball would have travelled a greater horizontal distance before striking the ground.
B The ball would have reached a greater maximum height.
C The ball’s horizontal velocity would have been continually decreasing.
9 A softball of mass 250 g is thrown with an initial velocity of 16 m s−1at an angle θto the horizontal. When the ball reaches its maximum height, its kinetic energy is 16 J.
a What is the maximum height achieved by the ball from its point of release?
b Calculate the initial vertical velocity of the ball.
c What is the value of θ?
d What is the speed of the ball after 1.0 s?
e What is the displacement of the ball after 1.0 s?
f How long after the ball is thrown will it return to the ground?
g Calculate the horizontal distance that the ball will travel during its flight.
10 During training, an aerial skier takes off from a ramp that is inclined at 40.0°to the horizontal and lands in a pool that is 10.0 m below the end of the ramp. If she takes 1.50 s to reach the highest point of her trajectory, calculate:
a the speed at which she leaves the ramp
b the maximum height above the end of the ramp that she reaches
c the time for which she is in mid-air.
2 0 m s–1
CHAPTER REVIEW For the following questions the
acceleration due to gravity may be taken as g=9.8 m s−2and the effects of air resistance can be ignored unless otherwise stated.
The following information applies to questions 1 and 2. The diagram shows the path taken by the centre of mass of a softball player sliding along the ground in a desperate attempt to make first base. The player is moving at a speed of 8.0 m s−1just before the slide and her speed 0.50 s later is 6.0 m s−1. The mass of the player is 80 kg.
1 a Calculate the magnitude of her change in velocity during this 0.50 s interval.
b Calculate her average acceleration during the 0.50 s interval.
2 What was the average frictional force acting on this player during the 0.50 s interval?
3 An Olympic archery competitor tests a bow by firing an arrow of mass 25 g vertically into the air.
The arrow leaves the bow with an initial vertical velocity of 100 m s−1. a At what time will the arrow
reach its maximum height?
b What is the maximum vertical distance that this arrow reaches?
c What is the acceleration of the arrow when it reaches its maximum height?
reaches her, it is travelling at 8.2 m s−1north.
a What is the velocity of the netball relative to Zahra?
b What is the velocity of Zahra relative to the netball?
5 An aeroplane is headed due north at 500 km h−1in still air. Then the wind starts to blow at a speed of 100 km h−1towards the west.
a What is the speed of the plane relative to the ground now?
b The pilot wishes to travel due north at 500 km h−1. In which direction and with what air speed should the pilot fly the plane to achieve this goal?
6 Two identical tennis balls X and Y are hit horizontally from a point 2.0 m above the ground with different initial speeds: ball X has an initial speed of 5.0 m s−1while ball Y has an initial speed of 7.5 m s−1.
a Calculate the time it takes for each ball to strike the ground.
b Calculate the speed of ball X just before it strikes the ground.
c What is the speed of ball Y just before it strikes the ground?
d How much further than ball X does ball Y travel in the horizontal direction before bouncing?
The following information relates to questions 7–10. The diagram shows the trajectory of a Vortex after it has been thrown with an initial speed of 10.0 m s−1. The Vortex reaches its maximum height at point Q; 4.00 m higher than its starting height.
7 What is the value of the angle θ that the initial velocity vector makes with the horizontal?
8 What is the speed of the Vortex at point Q?
9 What is the acceleration of the Vortex at point Q?
10 How far away is the Vortex when it reaches point R?
The following information applies to questions 11–15. In a shot-put event a 2.0 kg shot is launched from a height of 1.5 m, with an initial velocity of 8.0 m s−1at an angle of 60º to the horizontal.
11 a What is the initial horizontal speed of the shot?
b What is the initial vertical speed of the shot?
c How long does it take the shot-put to reach its maximum height?
d What is the maximum height from the ground that is reached by the shot?
e How long after being thrown does the shot reach the ground?
f Calculate the total horizontal distance that the shot travels during its flight.
12 What is the speed of the shot when it reaches its maximum height?
13 What is the minimum kinetic energy of the shot during its flight?
14 What is the acceleration of the shot at its maximum height?
15 Which of the following angles of launch will result in the shot travelling the greatest horizontal distance before returning to its initial height?
Questions 16–18 refer to the following information and diagram.
During a social soccer game, three of the players are running in different directions as shown in the following diagram. Jim, the goalkeeper, remains stationary.
16 What is the velocity of Jim relative to Sally?
17 What is the velocity of Sally relative to Harry?
18 Calculate the velocity of Harry relative to Maria.
The following information applies to questions 19 and 20. During an athletics meet, some physics students collected data at the shot-put event.
After analysing their data, some of the students managed to establish that during its flight the 2.0 kg shot experienced a force due to air resistance that was proportional to the square of its velocity. The formula R=3.78 ×10−5v2was suggested, where R is the force due to air resistance and v is the instantaneous speed of the shot. The shot-put in one particular toss was launched at 7.5 m s−1at an angle of 36°to the horizontal.
19 Calculate the maximum force due to air resistance that the shot experiences during its flight.
20 Calculate the value of the ratio of the forces acting on the shot as it is tossed:
What does your answer tell you about these forces?
Fg Fa(max) CHAPTER REVIEW
N
S
W E
7.5 m s–1 N
2.5 m s–1 NE
5.5 m s–1 E Maria
Harry
Sally
Jim