An NFL punter at the 15-yard line kicks a football with an initial velocity of 95 feet per second at an angle of elevation of 28 degrees. Let (t) be the elapsed time since the football is kicked.
a. Write the parametric equations of this problem.
b. With (t)=0 at the time of the kick, how many seconds is the ball in the air?
c. Where does the ball land given the kick is straight and from the 15 yard line?
a. To write the parametric equations for this problem, we need to break down the motion of the football into horizontal and vertical components.
Let's consider the horizontal motion first. The football will travel at a constant velocity in the horizontal direction. Since there are no forces acting horizontally (assuming no wind resistance), the horizontal velocity remains constant throughout the motion.
The horizontal component of the initial velocity can be calculated using the cosine of the angle of elevation. In this case, the initial velocity is 95 ft/s, and the angle of elevation is 28 degrees. So, the horizontal component of the initial velocity is given by:
Vx = initial velocity * cos(angle of elevation)
= 95 ft/s * cos(28 degrees)
Now let's consider the vertical motion. The football will experience the force of gravity, causing it to move in a parabolic trajectory. We need to consider the vertical displacement, initial vertical velocity, and the acceleration due to gravity.
The initial vertical velocity can be calculated using the sine of the angle of elevation. In this case, the initial velocity is 95 ft/s, and the angle of elevation is 28 degrees. So, the vertical component of the initial velocity is given by:
Vy = initial velocity * sin(angle of elevation)
= 95 ft/s * sin(28 degrees)
The vertical displacement at time (t) can be calculated using the equation:
y = initial vertical velocity * t - 0.5 * acceleration due to gravity * t^2
where the acceleration due to gravity is approximately 32.2 ft/s^2.
So, the parametric equations for this problem are:
x(t) = Vx * t
= 95 ft/s * cos(28 degrees) * t
y(t) = Vy * t - 0.5 * 32.2 ft/s^2 * t^2
= 95 ft/s * sin(28 degrees) * t - 0.5 * 32.2 ft/s^2 * t^2
b. To find the time the ball is in the air, we need to determine the time it takes for the football to hit the ground or reach a height of zero (y=0). We can set the equation for vertical displacement equal to zero and solve for (t):
0 = 95 ft/s * sin(28 degrees) * t - 0.5 * 32.2 ft/s^2 * t^2
This is a quadratic equation, and we can solve it by factoring, using the quadratic formula, or graphically.
Once you solve the quadratic equation, you will get two roots for (t), one positive and one negative. Since time cannot be negative in this context, you can disregard the negative root. The positive root will give you the time at which the ball hits the ground.
c. The horizontal distance traveled by the football can be calculated by multiplying the horizontal velocity (Vx) by the time (t) when the ball hits the ground. So the distance the ball lands from the kicker is:
Distance = Vx * t
Since you mentioned the kick was from the 15-yard line, you can add the initial distance (15 yards) to the calculated distance to find where the ball lands. Make sure to convert the units if necessary.