A football punter wants to kick the ball so that it is in...

A football punter wants to kick the ball so that it is in the air for 4.0 $\mathrm{s}$ and lands 50 $\mathrm{m}$ from where it was kicked. At what angle and with what initial speed should the ball be kicked? Assume that the ball leaves 1.0 $\mathrm{m}$ above the ground.

Key Concepts

Horizontal Range

The horizontal range refers to the total horizontal distance covered by the projectile. It is computed using the horizontal component of the initial velocity and the time of flight, making it a fundamental concept in determining the required initial speed for a desired distance.

Solving Quadratic Equations

When the launch point is not at ground level, solving the vertical displacement equation often leads to a quadratic equation. Being able to solve quadratic equations is essential in finding the time at which the projectile lands, taking into account the initial height, gravitational acceleration, and initial vertical velocity.

Time of Flight

The time of flight is the total duration an object remains in the air. It is primarily determined by the vertical component of the initial velocity and the acceleration due to gravity, taking into account any initial vertical displacement, and is key to predicting where and when the object will land.

Projectile Motion

Projectile motion describes the two-dimensional motion of an object under the influence of gravity, where horizontal and vertical motions are independent. This concept is essential when analyzing objects launched into the air because it allows separate treatment of the horizontal (constant velocity) and vertical (accelerated motion due to gravity) components.

Vector Decomposition

Any initial velocity in projectile motion can be broken down into horizontal and vertical components. This decomposition is crucial because the horizontal component determines the distance traveled while the vertical component influences the time of flight and maximum height reached.

Kinematic Equations

Kinematic equations govern the motion of objects under constant acceleration, such as gravity. They are used to connect displacement, initial velocity, acceleration, and time, and are essential in solving for unknown variables like time of flight and distance in projectile problems.

Transcript

00:01 Here in this given problem, the football leaves the ground from a point 1 .0 meter above the ground.

00:15 This is the angle of projection theta and initial velocity v0 both are missing.

00:22 Then the football goes like this and hit the ground at a distance 50 meter from the point of projection.

00:39 Time period of football in the air that is given as 4 .0 second and this range r that is 50 meter.

00:53 V0 that is missing, theta that is also missing.

01:00 Now there will be two components of this velocity, horizontal component vox and vertical component voy.

01:13 So this vox that is vo cos theta and voy it will be minus vo sin theta.

01:24 We are taking the direction upward to be negative.

01:29 So using second equation of motion because throughout its journey the vertical displacement covered by the football is just 1 .0 meter.

01:43 So using second equation of motion, displacement equation for the displacement of the football h is equal to voyt plus half gt square.

02:03 H that is 1 .0, voy that is minus vo sin theta multiplied by time which is time of flight 4 .0 second plus half 9 .8 multiplied by t square which is again 4.

02:22 So that is square of 4.

02:25 So the terms are rearranged to get 4 .0 sin theta is equal to 8 times 9 .8 minus 1 .0.

02:39 So this v0 here it should be v0 also.

02:49 So that is 4 .0 v0 sin theta...

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