An object that is in flight after being thrown or projected is called a projectile. Such a projectile might be a football, a cricket ball, a baseball or any other object. The motion of a projectile may be thought of as the result of two separate, simultaneously occurring components of motions. One component is along a horizontal direction without any acceleration and the other along the vertical direction with constant acceleration due to the force of gravity. It was Galileo who first stated this independency of the horizontal and the vertical components of projectile motion in his Dialogue on the great world systems (1632). In our discussion, we shall assume that the air resistance has negligible effect on the motion of the projectile. Suppose that the projectile is launched with velocity vo that makes an angle θo with the x-axis as shown in Fig. After the object has been projected, the acceleration acting on it is that due to gravity which is directed vertically downward.

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CASE_STUDIESProjectile Motion and Its AnalysisEquation of Trajectory of Projectile5 sub-questions

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Assertion: If there were no gravitational force, the path of the projected body always is a straight line. Reason: Gravitational force act along line joining centres of bodies.

Assertion-ReasonProjectile Motion and Its Analysis

When the football travels through the air, it always follows a parabolic path because the movement of the ball in the vertical direction is influenced by the force of gravity. As the ball travels up, gravity slows it down until it stops for a moment at its peak height; the ball then comes down and gravity accelerates it until it hits the ground. When a footballer kicks a ball he has to consider 3 factors: • The velocity at which the ball leaves his feet • The angle of kick • The rotation of the ball The rotation of the ball determines how the ball will slow down in flight. The velocity of the ball and the angle of the kick are the major factors those determine: • How long the ball will remain in air (hangtime) • How high the ball will go • How far will the ball go When the ball leaves the footballer's foot, it moves in two directions - horizontally and vertically. If the ball is kicked at a steep angle, then it will have more velocity in the vertical direction than in the horizontal direction - the ball will go high, have a long hang-time, but travel a short distance. If the ball is kicked at a shallow angle, it will have more velocity in the horizontal direction than in the vertical direction - the ball will not go very high, will have a short hang-time, but will travel a long distance. The footballer decides on the best angle in view of his field position. A footballer kicks a ball with velocity 60 m/s at an angle a. For which of the following values of a, the hang-time will be highest?

CASE_STUDIESProjectile Motion and Its AnalysisTime of Flight of ProjectileMaximum Height of Projectile