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.
Range = Vo^2*sin2A/g = 126 Ft.
98^2*sin2A/32 = 126
98*2*sin2A = 4032
sin2A = 0.42
2A = 24.8o
A = 12.4o.
Or A = 180 - 12.4 = 167.6o.
Range = Xo * T = 126 Ft.
98*cos12.4 * T = 126
T = 1.32 s. = Time in air.
Tr = T/2 = 1.32/2 = 0.658 s.=Rise time.
= Time at 12.4o.
Tf = Tr = 0.658 s. = Fall time. = Time
at 167.6o.
To determine the two possible values for the angle of the kick, theta, and the corresponding time at each angle, we can use the equations of projectile motion.
1. Calculate the vertical component of the initial velocity (v0y):
- The vertical component of the initial velocity can be found using the equation: v0y = v0 * sin(theta), where theta is the angle of the kick and v0 is the initial speed.
- Substitute the given value of v0 (98 ft/s) into the equation to get v0y in terms of theta.
2. Calculate the time it takes for the football to reach the highest point (t_max):
- At the highest point, the vertical component of the velocity becomes zero.
- Use the equation v_max = v0y - g * t_max, where v_max represents the maximum vertical component of the velocity, g is the acceleration due to gravity (32.2 ft/s^2), and t_max is the time taken to reach the highest point.
- Solve the equation for t_max.
3. Calculate the total time of flight (T):
- The total time of flight is the time it takes for the football to reach the ground again.
- Use the equation T = 2 * t_max to calculate the total time of flight.
4. Calculate the horizontal component of the initial velocity (v0x):
- The horizontal component of the initial velocity can be found using the equation v0x = v0 * cos(theta), where theta is the angle of the kick and v0 is the initial speed.
- Substitute the given value of v0 (98 ft/s) into the equation to get v0x in terms of theta.
5. Calculate the distance traveled by the football (d):
- The distance traveled by the football can be calculated using the equation d = v0x * T, where v0x is the horizontal component of the initial velocity and T is the total time of flight.
- Substitute the given value of d (126 ft) into the equation and solve for v0x * T.
6. Solve for the two possible values of theta:
- Using the two equations from steps 1 and 5, equate v0x * T to 126 ft to derive two separate equations for theta.
- Solve both equations for theta to find the two possible values of theta.
7. Calculate the corresponding time at each of the two angles (t1 and t2):
- Use the equation d = v0x * t1, where d is the distance traveled by the football (126 ft) and v0x is the horizontal component of the initial velocity.
- Solve the equation for t1.
- Repeat the same process to find t2 using the second value of theta.
By following these steps, you can determine the two possible values for the angle of the kick, theta, and the corresponding time at each angle.