The player kicks a football with an initial speed of v0 = 98 ft/s. Determine the two possible values for the angle of the kick, theta. The distance of the football field is 126 ft. Also determine the corresponding time at each of these two angles.
See previous post.
To determine the two possible values for the angle of the kick, theta, and the corresponding time at each of these angles, we can use the kinematic equations of motion. Let's break down the problem step by step:
1. First, let's identify the given information:
- Initial speed of the football (v0) = 98 ft/s
- Distance of the football field (d) = 126 ft
2. We need to find the two possible values for the angle of the kick, theta. To do this, we can use the range equation for projectile motion, which relates the angle, initial speed, and distance:
- Range (R) = v0^2 * sin(2θ) / g
Rearranging the equation, we can solve for the angle (θ):
- sin(2θ) = R * g / v0^2
- 2θ = sin^(-1)(R * g / v0^2)
- θ = 0.5 * sin^(-1)(R * g / v0^2)
3. Substituting the given values into the equation, we have:
- θ = 0.5 * sin^(-1)(126 * 9.8 / (98)^2)
Evaluating this using a calculator, we find two possible values for θ:
- θ1 ≈ 19.38 degrees
- θ2 ≈ 70.62 degrees
4. Now, to determine the corresponding time at each of these angles, we can use the equation for the time of flight (T) in projectile motion:
- T = 2 * v0 * sin(θ) / g
Substituting the given values, we have:
- T1 = 2 * 98 * sin(19.38) / 9.8
- T2 = 2 * 98 * sin(70.62) / 9.8
Evaluating these equations, we find the corresponding times:
- T1 ≈ 4.98 seconds
- T2 ≈ 11.43 seconds
Therefore, the two possible values for the angle of the kick, theta, are approximately 19.38 degrees and 70.62 degrees, with corresponding times of approximately 4.98 seconds and 11.43 seconds, respectively.