Typical Numerical-Shortcuts and Tricks

Trick- (Constant Accelration Equation): $$s=ut+\frac{1}{2}at^2$$ Where, ’s’ is the distance, ‘u’ is initial velocity, ’t’ is time, ‘a’ is the acceleration.

Trick- (Area Under V-T Graph):

• The area of the trapezium made by two points (t1,v1) and (t2,v2) on a v-t graph, represents the corresponding displacement between those two points. $$Area=\frac{1}{2}(v1+v2)(t2-t1)$$

Trick- (Projectile Motion):

• Range: The maximum horizontal distance covered by a projectile is called its range. In the absence of air resistance, the range of a projectile having an initial velocity ‘u’ and projection angle θ with the horizontal is given by $$R=\frac{u^2sin2\theta}{g}$$

• Maximum Height: The highest point reached by a projectile is called its maximum height. In the absence of air resistance, the maximum height attained by a projectile having an initial velocity ‘u’ and projection angle θ with the horizontal is given by $$H=\frac{u^2sin^2\theta}{2g}$$

• Time of Flight: The total time taken by a projectile to reach its maximum height and then return to the ground is called its time of flight. In the absence of air resistance, the time of flight of a projectile having an initial velocity ‘u’ and projection angle θ with the horizontal is given by $$T=\frac{2usin\theta}{g}$$

Trick- (Work-Energy Theorem)

• Work Done (W=F.S\cos\theta)
• If acceleration is constant and we have: $$v^2=u^2+2aS $$

Therefore, we can directly substitute acceleration with: $$a=\frac{v^2-u^2}{2s}$$ And hence calculate work done directly without calculating forces first.

Trick- (Circular Motion)

• When an object moves in a circular path, its velocity is constantly changing direction.
• The magnitude of the velocity remains the same, but the direction of the velocity vector is always tangent to the circular path.
• The acceleration of an object in circular motion is always directed toward the center of the circle.

Trick- (SHM)

• The period of oscillation of a mass-spring system is calculated as: $$T=2\pi\sqrt{\frac{m}{k}}$$

• The amplitude of oscillation represents maximum displacement from equilibrium position and can be calculated by (A=X_{max})

Trick- (Waves)

• The speed of a wave is equal to the product of its wavelength and frequency. $$ v=f\lambda$$
• For waves on a string, the tension in the string is equal to the product of the linear density of the string and the square of the wave speed. $$T=\mu v^2$$

These tricks can help you in quickly solving numerical problems related to motion, projectile motion, work, energy, and waves.