Motion in 2 Dimensions

Revision of basic formulae and concepts, useful for JEE/IAT and Boards.

In Standard Projectile Motion, What is the formula for total time of flight?

T=\dfrac{2u_y}{g}=\dfrac{2u\sin\theta}{g}

In Standard Projectile Motion, What is the formula for maximum height reached?

H=\dfrac{u_y²}{2g}=\dfrac{u²\sin²\theta}{2g}

In Standard Projectile Motion, What is the formula for Range of flight?

R=u_x\cdot T=\dfrac{2u_xu_y}{g}=\dfrac{u²\sin2\theta}{g}

What should the angle of projection be for maximum range in standard projectile motion?

\theta=45^\circ 

What is the formula for Maximum Range in Standard Projectile Motion?

R_{max}=\dfrac{u²}{g}

For same initial velocity, what is the condition for two angles of projection resulting in the same range?

\theta_1+\theta_2=90^\circ

What is the horizontal component of velocity of a projectile (showing standard projectile motion) at any time t?

v_x=u_x=u\cos\theta

What is the vertical component of velocity of a projectile (showing standard projectile motion) at any time t?

v_y=u_y-gt=u\sin\theta-gt  

What is the angle made from the horizontal by a projectile (showing standard projectile motion) at any time t?

\alpha=\tan^{-1}\Big(\dfrac{v_y}{v_x}\Big)=\tan^{-1}\Big(\dfrac{y}{x}\Big)

What is the equation of trajectory for standard projectile motion? (two of them)

y=x\tan\theta-\dfrac{gx²}{2u²\cos²\theta}=x\tan\theta-\dfrac{gx²}{2u_x²}=x\tan\theta\Big(1-\dfrac{x}{R}\Big)

If equation of trajectory for a standard projection motion is given as y=ax-bx², then what is the range of the projectile motion?

R=\dfrac{a}{b}

What is the formula for height of the tower a projectile is projected from, when time of flight and initial velocity are given?

-h=u_yT-\frac12gT²=u\sin\theta\ T-\frac12gT²   

What is the formula for range of projectile when it is projected from a tower, when time of flight is given?

R=u_xT=u\cos\theta\ T

When a projectile is projected horizontally from a tower, what is the formula for total time of flight?

T=\sqrt{\dfrac{2h}{g}}

When a projectile is projected horizontally from a tower, what is the formula for horizontal range of flight??

R=uT=u\sqrt{\dfrac{2h}{g}}

What is the key thing to remember when analyzing projectile motion on an inclined plane?

Both components of u will be retarded by g.

On a plain inclined at angle \theta, what is the formula for time of flight of a projectile projected at angle \alpha from the plane?

T=\dfrac{2u_y}{a_y}=\dfrac{2u\sin\alpha}{g\cos\theta}

On a plain inclined at angle \theta, what is the formula for maximum height (from the plane) of a projectile projected at angle \alpha from the plane?

H=\dfrac{u_y²}{2a_y}=\dfrac{u²\sin²\alpha}{2g\cos\theta}

On a plain inclined at angle \theta, what is the formula for Range (along the plan) of a projectile projected at angle \alpha from the plane?

R=u\cos\alpha\cdot T\pm\frac{1}{2}g\sin\theta\cdot T²

When the projectile is projected up the plane, use minus sign

When projectile is projected down the plain, use plus sign.

When a projectile is projected from a moving trolley (speed v_T at an angle \theta, What is the formula for the horizontal component of initial velocity of the projectile?

u_x=u\cos\theta+v_T

When a projectile is projected from a moving trolley (speed v_T at an angle \theta, What is the formula for the range velocity of the projectile?

R=(u\cos\theta+v_T)\cdot T

When a projectile is projected from a moving trolley (speed v_T at an angle \theta, What is the condition for maximum Range of the projectile?

\dfrac{dR}{d\theta}=0, then solve for maximisation