Projectile motion is a form of motion in which an object or particle moves along a curved path under the action of gravity only.
Assumptions of Projectile Motion
(1) There is no resistance due to air.
(2) The effect due to curvature of earth is negligible.
(3) The effect due to rotation of earth is negligible.
(4) For all points of the trajectory, the acceleration due to gravity 'g' is constant in magnitude and direction.
Principles of Physical Independence of Motions
Types of Projectile Motion
1. Oblique projectile motion: A projectile thrown with velocity u at an angle q with the horizontal. The velocity u can be resolved into two rectangular components. vcosq component along X-axis and usinq component along Y-axis.
Equation of trajectory:
For horizontal motion,
For vertical motion,
From equation (i) and (ii),
Time of flight:
For vertical upward motion,
Time taken to go up = The time taken to come down
Time of flight,
Horizontal range:
2. Horizontal Projectile: When a body is projected horizontally from a certain height 'h' with initial velocity u. If friction is considered to be absent, then there is no other horizontal force which can affect the horizontal motion. The horizontal velocity therefore remains constant and so the object covers equal distance in horizontal direction in equal intervals of time.
Trajectory of horizontal projectile:
For vertical motion,
For vertical motion,
From equation (iii) and (iv),
Time of flight:
Horizontal range: for horizontal motion
3. Projectile Motion on an Inclined Plane: When a particle is projected up with a speed u from an inclined plane which makes an angle a with the horizontal velocity of projection makes an angle q with the inclined plane.
Hence the component of initial velocity parallel and perpendicular to the plane are equal to ucosq and usinq respectively.
The component of g along the plane is gsina and perpendicular to the plane is gcosa as shown in the figure.
Therefore the particle decelerates at a rate of gsina as it moves from O to P.
Time of flight:
Horizontal range:
Horizontal range on an inclined plane,
Assumptions of Projectile Motion
(1) There is no resistance due to air.
(2) The effect due to curvature of earth is negligible.
(3) The effect due to rotation of earth is negligible.
(4) For all points of the trajectory, the acceleration due to gravity 'g' is constant in magnitude and direction.
Principles of Physical Independence of Motions
- The velocity of the particle can be resolved into two mutually perpendicular components. Horizontal component and vertical component.
- The horizontal component remains unchanged throughout the flight. The force of gravity continuously affects the vertical component.
Types of Projectile Motion
1. Oblique projectile motion: A projectile thrown with velocity u at an angle q with the horizontal. The velocity u can be resolved into two rectangular components. vcosq component along X-axis and usinq component along Y-axis.
Equation of trajectory:
For horizontal motion,
\[ x=utcos\theta \Rightarrow t=\frac{x}{ucos\theta } \cdot \cdot \cdot \cdot \cdot (i)\]
For vertical motion,
\[ y=(usin\theta )t-\frac{1}{2}gt^{2} \cdot \cdot \cdot \cdot \cdot \cdot (ii)\]
From equation (i) and (ii),
\[y=x tan\theta -\frac{1}{2}\frac{gx^2}{u^2cos^{2}\theta }\]
Time of flight:
For vertical upward motion,
\[0=usin\theta -gt\Rightarrow t=\frac{usin\theta }{g}\]
Time taken to go up = The time taken to come down
Time of flight,
\[T=2t=\frac{2usin\theta }{g}\]
Horizontal range:
\[R=T(ucos\theta )=\frac{u^{2}2sin\theta cos\theta}{g}=\frac{u^{2}sin2\theta }{g}\]
2. Horizontal Projectile: When a body is projected horizontally from a certain height 'h' with initial velocity u. If friction is considered to be absent, then there is no other horizontal force which can affect the horizontal motion. The horizontal velocity therefore remains constant and so the object covers equal distance in horizontal direction in equal intervals of time.
Trajectory of horizontal projectile:
For vertical motion,
\[x=ut\Rightarrow t=\frac{x}{u}\cdot \cdot \cdot \cdot \cdot (iii)\]
For vertical motion,
\[y=\frac{1}{2}gt^{2}\cdot \cdot \cdot \cdot \cdot (iv)\]
From equation (iii) and (iv),
\[y=\frac{1}{2}g\frac{x^2}{u^2}\]
Time of flight:
\[h=0+\frac{1}{2}gT^2\]
\[T=\sqrt{\frac{2h}{g}}\]
Horizontal range: for horizontal motion
\[R=uT+\frac{1}{2}0\times T^2\]
\[\Rightarrow R=uT=u\sqrt{\frac{2h}{g}}\]
3. Projectile Motion on an Inclined Plane: When a particle is projected up with a speed u from an inclined plane which makes an angle a with the horizontal velocity of projection makes an angle q with the inclined plane.
Hence the component of initial velocity parallel and perpendicular to the plane are equal to ucosq and usinq respectively.
The component of g along the plane is gsina and perpendicular to the plane is gcosa as shown in the figure.
Therefore the particle decelerates at a rate of gsina as it moves from O to P.
Time of flight:
\[T=\frac{2u_\perp }{a_\perp }=\frac{2usin\theta }{g cos\alpha }\]
(Analogous to oblique projectile)Horizontal range:
Horizontal range on an inclined plane,
\[R=u _{\parallel}T+\frac {1}{2}a_{\parallel}T^2\]
\[\Rightarrow R=ucos\theta \times T-\frac{1}{2}g sin\alpha \times T^2\]
By solving,
\[R=\frac{2u^{2}}{g}\frac{sin\theta\times cos(\theta +\alpha ) }{cos^{2}\alpha}\]
