A football is kicked at ground level with a speed of 20.7 m/s at an angle of 34.6° to the horizontal. How much later does it hit the ground?
The vertical velocity component, Voy, determines how long it takes to hit the ground.
Voy = 20.7 sin34.6 = 11.75 m/s
It hits the ground when t = 2 Voy/g
(That is, Voy/g to go up, and Voy/g to come back down from maximum height.)
Comp[ute 2 Voy/g
.417s
To find out how much later the football hits the ground, we need to calculate the time it takes for the ball to reach the ground. We can break down the velocity of the ball into its horizontal and vertical components.
The horizontal component of the velocity is given by:
Vx = V * cos(θ)
where V is the initial velocity of 20.7 m/s and θ is the angle of 34.6°.
Substituting the given values, we have:
Vx = 20.7 * cos(34.6°)
Similarly, the vertical component of the velocity is given by:
Vy = V * sin(θ)
where Vy is the vertical velocity.
Substituting the given values, we have:
Vy = 20.7 * sin(34.6°)
Since the ball is initially kicked at ground level, its initial vertical position is zero. We can use the equation of motion in the vertical direction to calculate the time it takes for the ball to hit the ground:
h = Vy * t + (1/2) * g * t^2
Since the ball hits the ground, h will be zero. And since the ball is going in the downward direction, g (acceleration due to gravity) will be negative.
0 = Vy * t - (1/2) * g * t^2
Substituting the values we have:
0 = 20.7 * sin(34.6°) * t - (1/2) * 9.8 * t^2
Simplifying the equation, we have:
0.5 * 9.8 * t^2 - 20.7 * sin(34.6°) * t = 0
This is a quadratic equation in terms of t. We can solve it by factoring, completing the square, or using the quadratic formula. Since the equation is already in standard form (ax^2 + bx + c = 0), we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 0.5 * 9.8, b = -20.7 * sin(34.6°), and c = 0. Substituting these values into the quadratic formula, we can calculate the two possible values of t. Since we want the positive value of t (time cannot be negative in this context), we only need to consider the positive square root:
t = (-(-20.7 * sin(34.6°)) ± √((-20.7 * sin(34.6°))^2 - 4 * 0.5 * 9.8 * 0)) / (2 * 0.5 * 9.8)
Calculating this expression will give us the time it takes for the ball to hit the ground.