Projectile Motion: Time to Change Direction

Question:

A particle is projected with a velocity u at an angle theta from the horizontal.After what time will it be perpendicular to the initial direction of projection?

Solution:

Projectile motion,examples and factors affecting:

Projectile motion-Image

Initial conditions:

The two components of the initial velocity u of the projectile are

• Horizontal component velocity $${u_x=u\cos\theta}$$
• Vertical component velocity $${u_y=u\sin\theta}$$

Projectile motion-motion along X-axis:

Projectile motion- motion along Y-axis:

Vertical components after time t:

• Horizontal component velocity (Remains constant) $${u_x=u\cos\theta}$$
• Vertical component velocity $${u_y=u\sin\theta-gt}$$

Vector expression for the initial velocity:

$${u\cos\theta\hat{\imath}+u\sin\theta\hat{\jmath}}$$

Vector expression for the velocity after time ‘t’

$${u\cos\theta\hat{\imath}+(u\sin\theta-gt)\hat{\jmath}}$$

Conditions for the two vectors to be perpendicular:

For the two vectors to be perpendicular, their dot product must be zero.The dot product of these two vectors is given by

$${[u\cos\theta\hat{\imath}+u\sin\theta\hat{\jmath}].[u\cos\theta\hat{\imath}+(u\sin\theta-gt)\hat{\jmath}]=0}$$

$${u^2\cos^2\theta+u^2\sin^2\theta-u\sin\theta gt=0}$$

$${u^2(\sin^2\theta+\cos^2\theta)-u\sin\theta gt}=0$$

$${u^2-u\sin\theta gt=0}$$

Time after which the velocity of the particle is perpendicular to its initial velocity:

$${t=\frac{u}{g\sin\theta}}$$

This is the time interval after which the velocity of the particle will be perpendicular to the initial velocity of the particle.

A PROJECTILE’S APPLICATIONS:

• Projectile ideas are used by football players to maximize kicks and shoots.
• Players in cricket employ projectile concepts to deliver and strike the ball with spin and speed. 
• The golf balls are propelled in a projectile manner.
• In baseball, for example. Pitch and trajectory are two further instances of projectile motion.
• In the sport of tennis. The serves and shots are based on the notion of projectile motion.
• Projectile motion is used to clearly describe physics principles.
• In experimentation and research, scientists employ projectiles to explore diverse phenomena.
• Rockets are launched as projectiles into space.
• Space probes are launched on precise paths to investigate celestial entities.
• Artillery, missile systems and mortar shells utilize projectile motion concepts.

PHYSICS QUIZ (PROJECTILE MOTION)

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