A jet plane flics horizontally at a hcight h at a speed v . An antiaircraft gun fires a shell at

A jet plane flics horizontally at a hcight h at a speed v ....

Key Concepts

Trigonometric Relationships

Trigonometric relationships, including functions like sine, cosine, and tangent, are used to break down the projectile’s initial velocity into horizontal and vertical components. These relationships help determine the angle of launch and relate the time of flight and range of the projectile to its motion, playing a critical role in solving for the conditions needed to intercept a target.

Optimization in Projectile Motion

Optimization in projectile motion involves determining the optimum conditions under which a projectile will achieve a specific objective, such as minimizing the initial speed required to hit a moving target. This usually requires setting up the motion equations, applying constraints, and then using calculus or algebraic methods to find the critical values for variables such as launch speed and angle.

Relative Motion

Relative motion involves analyzing the movement of one object in relation to another. In the context of hitting a moving target, such as a jet in flight, it is necessary to account for the target’s motion when solving for the projectile's path. This concept allows for establishing the relationship between the projectile’s trajectory and the target’s position over time.

Projectile Motion

Projectile motion is the study of an object that is launched into the air and continues its movement under the influence of gravity alone, without propulsion. Its path, typically a parabola, is determined by the initial velocity, launch angle, and gravitational acceleration. Understanding projectile motion is essential for deriving the trajectory equations needed to predict where and when the projectile will intersect with a moving target.

Kinematic Equations

Kinematic equations are the mathematical tools used to describe the motion of objects with constant acceleration. They relate displacement, velocity, acceleration, and time. For solving problems like intercepting a moving plane, these equations are separated into horizontal and vertical components to match the projectile's motion with the target's trajectory.

Transcript

00:01 Hello students in this question we have to approve that it hit the plane when the velocity is under root of v square 12g at an angle 10 inverse of under root of 2g h upon v okay so first of all i will draw the diagram so in this direction we will have v is equal to u cos alpha right and this bullet will go something like this and this is the point where it is fired and this is the jet plane and here it will cross the jet plane.

00:33 So here it will be hated by jet plane.

00:35 So now this height is h and in this direction we will in this direction we will have u.

00:43 So in the vertical direction we will have u -sine alpha.

00:46 Okay so now kinetic energy.

00:50 So we know that s is equal to u -t plus half a t square right so this is half a t square okay so now we know that this s will be equal to u and this u will be in the vertical direction that is sine alpha and to t minus half of acceleration will be g okay t square so from here i get an equation that is half gt square minus u sine alpha t plus h is equal to zero so now uh we know that since t is real so we can say that b of this equation that is b square minus 4 ac will be greater than 0.

01:36 So from here i say that u square sine square alpha minus 4 into half of g of h.

01:46 This will be greater than 0.

01:49 So from here i get u square sine square alpha is greater than 2g h okay and similarly we know that in the horizontal direction we have u course alpha so so we can say that the angle, this h would be under root of u square minus v square, that is the vertical direction.

02:20 So in this equation, if i put at the place of sine alpha, i would get u square, u square minus v square upon u square is greater than 2gh.

02:33 So from here what do i get is u square is greater than 2gh plus v plus v square...

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