5Projectile Motion

📚
PROJECTILE MOTION
▢ Definition:

Table 1: Projectile Motion

Term
Meaning
Condition
Initial velocity and acceleration act at angle other than 0° or 180°
Path
Parabolic
Projectile motion
Motion of particle in plane under gravity alone
Projectile
Particle thrown into space
Acceleration
Constant = \(g\) downward
Velocity
Changes in magnitude and direction
▢ Types:

Table 1: Types of Projectile Motion

Type
Condition
Horizontal projectile
Particle projected horizontally from height \(H\) with speed \(u\)
Oblique projectile
Particle projected from ground with velocity \(u\) at angle \(\theta\) with horizontal
▢ Horizontal Projectile from Height H:
❖ Condition: Projected horizontally from height \(H\) with speed \(u\)
❖ Components:

Table 1: Horizontal Projection Components

Component
Value
Horizontal velocity
\(u_x=u\)
Vertical velocity initially
\(u_y=0\)
Horizontal acceleration
\(a_x=0\)
Vertical acceleration
\(a_y=-g\)
❖ Motion Equations:

Table 1: Horizontal Projectile Formulae

Quantity
Formula
Horizontal distance in time \(t\)
\(x=ut\)
Vertical position in time \(t\)
\(y=H-\frac{1}{2}gt^2\)
Trajectory
\(y=H-\frac{1}{2}g\left(\frac{x}{u}\right)^2\)
Time of flight
\(T=\sqrt{\frac{2H}{g}}\)
Horizontal range
\(R=uT=u\sqrt{\frac{2H}{g}}\)
Vertical range
\(H\)
Final vertical velocity
\(v_y=gT=\sqrt{2gH}\)
Final velocity
\(v_f=\sqrt{u^2+(gT)^2}=\sqrt{u^2+2gH}\)
Angle with horizontal at striking point
\(\tan\phi=\frac{v_y}{v_x}=\frac{gT}{u}=\frac{\sqrt{2gH}}{u}\)
❖ Energy and Direction:
  • •
    Velocity, speed, kinetic energy and momentum increase gradually
  • •
    Potential energy gradually decreases
  • •
    Mechanical energy remains constant
  • •
    Acceleration remains constant
  • •
    Angle between velocity and acceleration decreases gradually
▢ Oblique Projection:
❖ Condition: Projected from ground with initial velocity \(u\) at angle \(\theta\) with horizontal
❖ Components:

Table 1: Oblique Projection Components

Component
Value
Horizontal velocity
\(u_x=u\cos\theta\)
Vertical velocity
\(u_y=u\sin\theta\)
Horizontal acceleration
\(a_x=0\)
Vertical acceleration
\(a_y=-g\)
❖ Motion Equations:

Table 1: Oblique Projectile Formulae

Quantity
Formula
Horizontal distance in time \(t\)
\(x=u\cos\theta\ t\)
Vertical distance in time \(t\)
\(y=u\sin\theta\ t-\frac{1}{2}gt^2\)
Trajectory
\(y=x\tan\theta-\frac{g x^2}{2u^2\cos^2\theta}\)
Time of flight
\(T=\frac{2u\sin\theta}{g}\)
Horizontal range
\(R=\frac{u^2\sin2\theta}{g}\)
Maximum height
\(H=\frac{u^2\sin^2\theta}{2g}\)
Useful relation
\(\frac{1}{2}gT^2=R\tan\theta=4H\)
For \(R=H\)
\(\theta=\tan^{-1}(4)=75.96^\circ\)
❖ Velocity:

Table 1: Velocity During Oblique Motion

Point
Velocity
At projection
\(u\)
At highest point
\(u\cos\theta\)
At same level of projection
\(u\)
At highest point vertical velocity
\(v_y=0\)
At highest point horizontal velocity
\(v_x=u\cos\theta\)
At ground, angle of velocity
\(\theta\) below horizontal
❖ Energy and Direction:
  • •
    Speed, velocity, momentum and kinetic energy first decrease
  • •
    At highest point, speed/KE/momentum become minimum
  • •
    After highest point, speed/KE/momentum increase
  • •
    Potential energy first increases, becomes maximum at highest point, then decreases
  • •
    Mechanical energy remains constant
  • •
    Acceleration remains constant
  • •
    Angle between acceleration and velocity changes from \((90^\circ+\theta)\) to \((90^\circ-\theta)\)
  • •
    At highest point, velocity and acceleration are perpendicular
▢ At Highest Point of Oblique Projectile:
❖ Given: Mass \(m\), initial velocity \(u\), angle \(\theta\), initial KE \(E_0\), initial momentum \(p_0\)

Table 1: Quantities at Highest Point

Quantity
Formula
Horizontal velocity
\(v_x=u\cos\theta\)
Vertical velocity
\(v_y=0\)
Net velocity
\(v=u\cos\theta\)
Momentum
\(P=mu\cos\theta=p_0\cos\theta\)
Kinetic energy
\(E_K=\frac{1}{2}mv^2=E_0\cos^2\theta\)
Potential energy
\(E_P=mgH=E_0\sin^2\theta\)
Total mechanical energy
\(E_0\)
▢ Change in Quantities:

Table 1: Change from Projection

Quantity
From departure to highest point
From departure to arrival
Speed
\(u-u\cos\theta=u(1-\cos\theta)=2u\sin^2\frac{\theta}{2}\)
0
Velocity
\(\Delta u_x=0,\ \Delta u_y=u\sin\theta,\ \Delta u=u\sin\theta\)
\(2u\sin\theta\)
Momentum
\(mu\sin\theta\)
\(2mu\sin\theta=2p_0\sin\theta\)
Kinetic energy
\(E_0-E_0\cos^2\theta=E_0\sin^2\theta\)
0
Potential energy
\(E_0\sin^2\theta\)
0
Direction change
\(\theta\)
\(2\theta\)
Average velocity
\(\frac{u}{2}\sqrt{1+3\cos^2\theta}\)
\(u\cos\theta\)
▢ Effect of Air Resistance:
  • •
    Maximum height decreases
  • •
    Horizontal range decreases
  • •
    Speed at striking point decreases
  • •
    Momentum at striking point decreases
  • •
    Kinetic energy at striking point decreases
  • •
    Time of flight is greater when air resistance is neglected
  • •
    With air resistance, angle of striking is greater than angle of projection
▢ Read and Digest:

Table 1: Projectile Special Points

Case
Result
To hit a target with rifle
Aim slightly higher than target
Body dropped from uniformly moving vehicle
Path appears straight line to passenger
Same dropped body seen from ground
Path appears parabolic
One ball falls vertically and another is projected horizontally from same height
Both reach ground simultaneously
Time to reach ground from height \(h\)
\(t=\sqrt{\frac{2h}{g}}\)
One ball dropped and another thrown horizontally from same height
Horizontally thrown ball reaches ground with greater speed
Javelin throw
Best practical angle is slightly less than 45°
▢ High-Yield Recall:

Table 1: Projectile Motion One-Liners

Fact
Answer
Path of projectile
Parabola
Projectile acceleration
\(g\) downward
Horizontal acceleration
0
Vertical acceleration
\(-g\)
Horizontal projectile time
\(T=\sqrt{\frac{2H}{g}}\)
Horizontal projectile range
\(R=u\sqrt{\frac{2H}{g}}\)
Horizontal projectile final velocity
\(v_f=\sqrt{u^2+2gH}\)
Oblique projectile time of flight
\(T=\frac{2u\sin\theta}{g}\)
Oblique projectile range
\(R=\frac{u^2\sin2\theta}{g}\)
Maximum height
\(H=\frac{u^2\sin^2\theta}{2g}\)
Range maximum angle
\(45^\circ\)
Velocity at highest point
\(u\cos\theta\)
Vertical velocity at highest point
0
KE at highest point
\(E_0\cos^2\theta\)
PE at highest point
\(E_0\sin^2\theta\)
Momentum at highest point
\(p_0\cos\theta\)
Mechanical energy
Constant
If \(R=H\)
\(\theta=\tan^{-1}4\)
Passenger view of dropped body
Straight line
Ground observer view of dropped body
Parabola
Javelin practical angle
Slightly less than 45°
Q1.
When air resistance is taken into account\nwhile dealing with the motion of the\nprojectile. Of the following properties of\nthe projectile, the one which shows an\nincrease is:\n[BP 2014)
📅BP 2014
Q2.
A body is thrown with velocity 50 m/s. The\nmaximum horizontal distance it can cover\n[BP 2010]
📅BP 2010
Q3.
The range of a projectile fired at an angle\nof 15" is 50 m. If it is fired with the same\nspeed at an angle of 45", its range will be;
Q4.
In a projectile throw at the same angles the\nvelocity of A is two times of velocity of B\ntheir ranges are related as\n[IOM 2014]
📅IOM 2014
Q5.
A body is projected at an angle 45" with \na velocity 200 m/s. The maximum height\nattained by the projectile is\n[MOE 2012,11]
📅MOE 2012•MOE 2011
Q6.
A body projected at angle of 60" attains the\nsame height at two points a and b at two\ndifferent times. If air resistance is\nneglected the ratio of mechanical energies\nat with respect to that at b will be:\n\n(MOE 2011]
📅MOE 2011
Q7.
A fielder can throw a cricket ball to a\nmaximum horizontal distance of 100 m.\nHow high the fielder can throw the same\nball?\n[IOM 2013)
📅IOM 2013
Q8.
A stone is projected horizontally at 30m/s\nfrom the top of a tower 45m high. At what\nangle with horizontal, the stone strikes the\nhorizontal ground at the same level of the\nfoot of the tower?
Q9.
If maximum height of a projectile is\nincreased by 10% keeping 0 same, then the\ntime of flight increases by
Q10.
10. A stone is thrown horizontally forward at\n30m/s from the top of a tower 45m high.\nThen displacement of the stone from the\nfoot of the tower after one second is
Q11.
11. A projectile is thrown at angle 0 with\nvertical with initial kinetic energy Eo- If air\nresistance is neglected the K.E. at highest\npoint will be\n[MOE 2013]
📅MOE 2013
Q12.
12. It is possible to project a particle with a\ngiven velocity in two possible ways so as to\nmake it pass through a point at a distance\nr from the point of projection. The product\nof times taken to reach this point in the\ntwo possible ways is then proportional to
Q13.
13. From the top of a tower of height 40m a\nball is projected upwards with a speed of\n20m/sec at an angle of elevation of 30.\nThen the ratio of the total time taken by\nthe ball to hit the ground to its time of\nflight is (g = 10m/s')
Q14.
14. A gun fires two bullets at 60' and 30' with\nhorizontal. The bullets strike at some\nhorizontal distance. The ratio of maximum\nheight for the two bullets is in the ratio
Q15.
15. The maximum range of a gun on\nhorizontal train is 16km. If g is 10m/s', the\nmuzzle velocity of the shell will be
Q16.
16. A javelin thrown into air at an angle with\nthe horizontal has a range of 200m. If the\ntime of flight is 5 second, then the\nhorizontal component of velocity of the\nprojectile at the highest point of the\ntrajectory is
Q17.
17. A ball as projected upwards from the top\n\nof tower with vertical velocity 50m/s\nmaking an angle 30" with the horizontal.\nThe height of the tower is 70m. After how\nmany seconds from the instant of throwing\nwill the ball reach the ground?
Q18.
18. The equation of trajectory from a\nprojectile thrown horizontally from\ncertain height above earth is y = 45 - 20\nwhere x and y are in metre. Then initial\nhorizontal velocity is
Q19.
19. From the above question, find the\nhorizontal range?
Q20.
20. A projectile thrown with speed u at angle 0\nwith horizontal is moving at right angle to\nits initial direction when its velocity is\nutane\nusine\nc. ucote\nusece
Q21.
21. For what angle of projection with\nhorizontal, the horizontal range of a\nprojectile is equal to its maximum vertical\nheight?
Q22.
22. A projectile thrown at certain angle with\nhorizontal from horizontal ground is\n\nmoving horizontally after 2 seconds and is\nmoving at 45" after one second. Then the\nangle of projection for the projectile is
Q23.
23. A ball is thrown vertically upward at\ninitial velocity 20m/s from roof of a bus\ntravelling along constant velocity 30m/s\nalong a straight road. The ball returns to\nthrower's hand after the bus has travelled
Q24.
24. A body is projected with velocity v1 from\nthe point A as shown in the figure. At the\nsame time, another body is projected\n\nvertically upwards from B with velocity v2.\nThe point B lies vertically below the\nhighest point. For both the bodies to\ncollide, - should be
Q25.
25. The equation of the trajectory of an\noblique projectile is y = V3x - -gx\nHere x and y are in metre and g is in m/s'.\nThe angle of projectile is
Q26.
26. A projectile is fired with velocity u making\nangle 0 with the horizontal. What is the\n\nangular momentum of the projectile at the\nhighest point about the starting point?\nGiven the mass of the projectile is m
Q27.
27. Two projectile are fixed from the same\npoint with the same speed at angles of\nprojection 60"and 30 respectively. Which\none of the following is true?
Q28.
28. The range of a projectile when projected at\nan angle 0 degree is R. If the angle of\nprojection is 20 degree but the range\nremains the same the angle will be\n[MOE 2009]
Q29.
29.\nA projectile's time of flight 'T' is related to\nhorizontal range by equation gT = 2R\nThe angle of projection in degrees is:\n[TOM 063]
Q30.
30. A ball is thrown horizontally from the\ntop of a tower. Horizontal component of\nits velocity will\n[MOE 066]
📅MOE 066
Q31.
31. Two bullets A and B are fired horizontally\nat the same instant of time with velocity\nV. and V, from the same height. If V. is\ngreater than V, which will reach the\nground first?\n[MOE 2065]
📅MOE 2065
Q32.
32.\nA cricket ball is struck with a K.E of'K' at\nan angle of 45' with\nthe horizontal. The\nat the maximum height will be.\n[MOE 20621
Q33.
K is the horizontal range of projectile for\nan angle of projection of 150. For the same\nrange an another angle of projection will\nbe\n[MOE 2000]
📅MOE 2000
Q34.
34. For a projectile fired at equal inclination to\nhorizontal and vertical line with velocity u,\nthe horizontal distance travelled is: [MOE]
Q35.
35. Two projectile A & B are thrown at a sam\nangle with velocity us = Zu, then what is\nthe relation between their range [TE-02]
Q36.
36. If mass & velocity of a body in a projectile\nmotion is doubled then linear momentum\nbecomes\nLIE-05
Q37.
37. Ball A is dropped vertically B is thrown\nhorizontally from the same height and a\nthe same moment, then:\n[IE-071
Q38.
38. The greatest height to which man can\nthrow a stone is h. The greatest distance to\nwhich he can throw it will be [BPKIHS-09]
📅BPKIHS-09
Q39.
39. If the maximum range of a projectile is\n\nfour times of its maximum height, then the\nangle of projection is equal to [ BPKIHS-07]
📅BPKIHS-07
Q40.
40. A plane is moving with 300ms" when it is\n\njust above the observer at a height of 2km.\nAt what angle with the vertical must he\nfire the gun with velocity 400 m/sec\ndirectly to hit the aeroplane? [BPKIHS 02]
📅BPKIHS 02
Q41.
41. A man in a train moving with a constant\nvelocity drops a ball on the platform. The\npath of the ball as seen by an observer\n[KU 2015]\nstanding on the platform is
📅KU 2015
Q42.
42. The range of projectile fired at an angle of\n15 is 50 m. If it is fired with same speed at\nan angle of 450 then, range will be:\n[KU 2017]
📅KU 2017
Q43.
43. A stone is released from a top of tower.\n\nWhen a constant force is applied by high\nspeed wind on that stone, then the path it\n[IOM 2017]\nfollows is:
📅IOM 2017