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 heta 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- rac{1}{2}gt^2
Trajectory
y=H- rac{1}{2}g eft( rac{x}{u} ight)^2
Time of flight
T= qrt{ rac{2H}{g}}
Horizontal range
R=uT=u qrt{ rac{2H}{g}}
Vertical range
H
Final vertical velocity
v_y=gT= qrt{2gH}
Final velocity
v_f= qrt{u^2+(gT)^2}= qrt{u^2+2gH}
Angle with horizontal at striking point
an hi= rac{v_y}{v_x}= rac{gT}{u}= rac{ qrt{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 heta with horizontal
❖ Components:

Table 1: Oblique Projection Components

Component
Value
Horizontal velocity
u_x=u os heta
Vertical velocity
u_y=u in heta
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 os heta t
Vertical distance in time t
y=u in heta t- rac{1}{2}gt^2
Trajectory
y=x an heta- rac{g x^2}{2u^2 os^2 heta}
Time of flight
T= rac{2u in heta}{g}
Horizontal range
R= rac{u^2 in2 heta}{g}
Maximum height
H= rac{u^2 in^2 heta}{2g}
Useful relation
rac{1}{2}gT^2=R an heta=4H
For R=H
heta= an^{-1}(4)=75.96^ irc
❖ Velocity:

Table 1: Velocity During Oblique Motion

Point
Velocity
At projection
u
At highest point
u os heta
At same level of projection
u
At highest point vertical velocity
v_y=0
At highest point horizontal velocity
v_x=u os heta
At ground, angle of velocity
heta 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^ irc+ heta) to (90^ irc- heta)
  • •
    At highest point, velocity and acceleration are perpendicular
▢ At Highest Point of Oblique Projectile:
❖ Given: Mass m , initial velocity u , angle heta , initial KE E_0 , initial momentum p_0

Table 1: Quantities at Highest Point

Quantity
Formula
Horizontal velocity
v_x=u os heta
Vertical velocity
v_y=0
Net velocity
v=u os heta
Momentum
P=mu os heta=p_0 os heta
Kinetic energy
E_K= rac{1}{2}mv^2=E_0 os^2 heta
Potential energy
E_P=mgH=E_0 in^2 heta
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 os heta=u(1- os heta)=2u in^2 rac{ heta}{2}
0
Velocity
elta u_x=0, elta u_y=u in heta, elta u=u in heta
2u in heta
Momentum
mu in heta
2mu in heta=2p_0 in heta
Kinetic energy
E_0-E_0 os^2 heta=E_0 in^2 heta
0
Potential energy
E_0 in^2 heta
0
Direction change
heta
2 heta
Average velocity
rac{u}{2} qrt{1+3 os^2 heta}
u os heta
▢ 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= qrt{ rac{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= qrt{ rac{2H}{g}}
Horizontal projectile range
R=u qrt{ rac{2H}{g}}
Horizontal projectile final velocity
v_f= qrt{u^2+2gH}
Oblique projectile time of flight
T= rac{2u in heta}{g}
Oblique projectile range
R= rac{u^2 in2 heta}{g}
Maximum height
H= rac{u^2 in^2 heta}{2g}
Range maximum angle
45^ irc
Velocity at highest point
u os heta
Vertical velocity at highest point
0
KE at highest point
E_0 os^2 heta
PE at highest point
E_0 in^2 heta
Momentum at highest point
p_0 os heta
Mechanical energy
Constant
If R=H
heta= an^{-1}4
Passenger view of dropped body
Straight line
Ground observer view of dropped body
Parabola
Javelin practical angle
Slightly less than 45°
Q1.
A body is projected making angle 60° with vertical. What is the value of KE at the highest point?
📅CEE MBBS 2024B
Q2.
If the velocity of a body at maximum height is half of the initial velocity, find the range of the projectile if the initial velocity is u.
📅CEE MBBS 2024a
Q3.
Two bodies are projected at angles 45° and 60° respectively with the horizontal. If both attain the same maximum height, the ratio of their ranges is:
📅CEE MBBS 2021
Q4.
What is the ratio of the time of flight when a projectile is projected at an angle of θ and (90 − θ)?
📅CEE BDS 2024
Q5.
In a projectile motion, maximum horizontal range is obtained at the projection angle of ..... with horizontal:
📅CEE BDS 2024
Q6.
A plane is moving with 300 ms⁻¹ when it is just above the observer at a height of 2 km. At what angle with the vertical must he fire the gun with velocity 600 m/s directly to hit the aeroplane?
📅CEE BDS 2024
Q7.
The total time of flight for an object thrown upward with initial velocity u and g gravity of planet is:
📅CEE BDS 2023
Q8.
A fielder can throw a cricket ball to a maximum horizontal distance of 100 m. How high the fielder can throw the same ball?
📅CEE BDS 2021
Q9.
When air resistance is included, which property of a projectile generally increases relative to motion in vacuum?
📅BP 2014
Q10.
A projectile is launched with speed \(50\,\mathrm{m\,s^{-1}}\). Find its maximum horizontal range for \(g=10\,\mathrm{m\,s^{-2}}\).
📅BP 2010
Q11.
A projectile has a range of \(50\,\mathrm{m}\) at \(15^\circ\). What is its range at \(45^\circ\) with the same launch speed?
Q12.
Two projectiles are launched at the same angle with \(u_A=2u_B\). Compare their horizontal ranges.
📅IOM 2014
Q13.
A projectile is launched at \(45^\circ\) with speed \(200\,\mathrm{m\,s^{-1}}\). Calculate its maximum height if \(g=10\,\mathrm{m\,s^{-2}}\).
📅MOE 2012•MOE 2011
Q14.
A projectile passes through points \(A\) and \(B\) at equal heights, at different times. With no air resistance, what is the ratio of its mechanical energies \(E_A/E_B\)?
📅MOE 2011
Q15.
A fielder can throw a ball to a maximum horizontal range of \(100\,\mathrm{m}\). How high can the ball rise if thrown vertically upward with the same speed?
📅IOM 2013
Q16.
A stone is projected horizontally at \(30\,\mathrm{m\,s^{-1}}\) from a tower \(45\,\mathrm{m}\) high. Find its impact angle below the horizontal for \(g=10\,\mathrm{m\,s^{-2}}\).
Q17.
Keeping the projection angle unchanged, the maximum height of a projectile increases by \(10\%\). By approximately what percentage does its time of flight increase?
Q18.
A stone is thrown horizontally at \(30\,\mathrm{m\,s^{-1}}\) from a tower \(45\,\mathrm{m}\) high. What is its straight-line distance from the foot of the tower after \(1\,\mathrm{s}\)? Take \(g=10\,\mathrm{m\,s^{-2}}\).
Q19.
A projectile is launched at angle \(\theta\) **with the vertical**, with initial kinetic energy \(E_0\). Neglecting drag, its kinetic energy at the highest point is:
📅MOE 2013
Q20.
A projectile launched with fixed speed \(u\) lands at the same ground-level point a horizontal distance \(r\) away for two complementary launch angles. The product of their flight times is proportional to:
Q21.
A ball is launched from a \(40\,\mathrm{m}\) tower at \(20\,\mathrm{m\,s^{-1}}\), \(30^\circ\) above horizontal. Find the ratio of time to hit the ground to the flight time for return to launch level. Take \(g=10\,\mathrm{m\,s^{-2}}\).
Q22.
Two bullets are fired with the same speed at \(60^\circ\) and \(30^\circ\) to the horizontal. Find the ratio of their maximum heights \(H_{60}:H_{30}\).
Q23.
The maximum horizontal range of a gun is \(16\,\mathrm{km}\). Find its muzzle speed, taking \(g=10\,\mathrm{m\,s^{-2}}\).
Q24.
A javelin has horizontal range \(200\,\mathrm{m}\) and flight time \(5\,\mathrm{s}\). Its horizontal speed at the highest point is:
Q25.
A ball is launched at \(50\,\mathrm{m\,s^{-1}}\), \(30^\circ\) above horizontal, from a tower \(70\,\mathrm{m}\) tall. When does it hit the ground? Use \(g=10\,\mathrm{m\,s^{-2}}\).
Q26.
A projectile launched horizontally follows \(y=45-\frac{x^2}{20}\), with \(x,y\) in metres. What is its initial horizontal speed if \(g=10\,\mathrm{m\,s^{-2}}\)?
Q27.
For a horizontal projectile with trajectory \(y=45-\frac{x^2}{20}\) (metres), find the horizontal range to ground level \(y=0\).
Q28.
A projectile is launched at speed \(u\), angle \(\theta\) above the horizontal. What is its speed when its velocity is perpendicular to its initial velocity?
Q29.
At what projection angle \(\theta\) does horizontal range equal maximum height for a projectile launched and landing at the same level?
Q30.
A projectile moves horizontally at \(t=2\,\mathrm{s}\) and its velocity makes \(45^\circ\) above the horizontal at \(t=1\,\mathrm{s}\). Find the launch angle.
Q31.
A ball is thrown vertically upward at \(20\,\mathrm{m\,s^{-1}}\) inside a bus moving uniformly at \(30\,\mathrm{m\,s^{-1}}\). How far does the bus move before the ball returns to the thrower? Assume \(g=10\,\mathrm{m\,s^{-2}}\).
Q32.
The trajectory of a projectile is \(y=\sqrt3\,x-kx^2\), where \(k>0\). Find its launch angle above the horizontal.
Q33.
A projectile of mass \(m\) is launched with speed \(u\) at angle \(\theta\) above horizontal. Find the **magnitude** of its angular momentum about the launch point at maximum height.
Q34.
Two projectiles are launched with identical speeds from level ground at \(30^\circ\) and \(60^\circ\). Which statement is true?
Q35.
At a fixed launch speed, a projectile has the same range at \(20^\circ\) and at what other angle?
📅MOE 2009
Q36.
For a projectile launched and landing at the same level, \(gT^2=2R\), where \(T\) is its flight time and \(R\) its range. Find the projection angle.
📅IOM 2063 (source printed TOM 063)
Q37.
A ball is projected horizontally from a tower. Neglecting air resistance, its horizontal velocity component:
📅MOE 066
Q38.
Two bullets are fired horizontally from the same height at the same time, with different horizontal speeds. Which reaches the ground first?
📅MOE 2065
Q39.
A cricket ball is projected at \(45^\circ\) with initial kinetic energy \(K\). Find its kinetic energy at maximum height.
📅MOE 2062
Q40.
A projectile has a given range at \(15^\circ\). At what other launch angle does it have the same range with unchanged speed?
📅MOE 2000
Q41.
A projectile is fired at equal inclination to the horizontal and vertical with speed \(u\). Find its horizontal range.
📅MOE (year unspecified)
Q42.
Two projectiles launch at the same angle with \(u_B=2u_A\). What is the relation between ranges?
📅IE 2002
Q43.
If both the mass and velocity magnitude of a projectile are doubled, its linear momentum magnitude becomes:
📅IE 2005
Q44.
Ball \(A\) is dropped vertically and ball \(B\) is projected horizontally with nonzero speed from the same height at the same instant. Neglecting air resistance, which statement is correct?
📅IE 2007
Q45.
The greatest height to which a person can throw a stone is \(h\). With the same speed, what is its greatest horizontal range?
📅BPKIHS-09
Q46.
For a particular projectile, its horizontal range is four times its maximum height. What is its launch angle?
📅BPKIHS-07
Q47.
A plane moves horizontally at \(300\,\mathrm{m\,s^{-1}}\). When it is overhead, a projectile is fired at speed \(400\,\mathrm{m\,s^{-1}}\). Ignoring gravity for this **direction-only** interception approximation, find the angle with the vertical required for matching horizontal speed.
📅BPKIHS 02
Q48.
A passenger on a train moving at constant velocity drops a ball. What path does a stationary observer on the platform see, neglecting air resistance?
📅KU 2015
Q49.
A projectile has range \(50\,\mathrm{m}\) at \(15^\circ\). What is its range at \(45^\circ\) with the same speed?
📅KU 2017
Q50.
A stone is released from rest from a tower and experiences a constant horizontal force from wind throughout its fall. What is its trajectory if the net acceleration is constant?
📅IOM 2017