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