A golfer, standing on a fairway, hits a shot to a green that is elevated 5.80 m above the point where she is standing. If the ball leaves her club with a velocity of 46.3 m/s at an angle of 36.0° above the ground, find the time that the ball is in the air before it hits the green.
Vo = 46.3m/s[36o].
Xo = 46.3*cos36 = 37.5 m/s.
Yo = 46.3*sin36 = 27.21 m/s.
h = (Y^2-Yo^2)/2g = (0-(27.21^2))/-19.6=
37.8 m. Above gnd.
Y = Yo + g*Tr = 0
Tr = -Yo/g = -27.21/-9.8 = 2.78 s. =
Rise time.
d = 0.5g*t^2 = 37.8-5.8 = 32 m. to fall
4.9t^2 = 32
t^2 = 6.53
Tf = 2.56 s
Tr+Tf = 2.78 + 2.56 = 5.34 s. = Time in
air.
To find the time that the ball is in the air before hitting the green, we can use the projectile motion equations.
The first step is to break down the initial velocity of the ball into its vertical and horizontal components.
The vertical component of the velocity can be found using the equation: Vy = V * sin(theta), where V is the initial velocity of the ball (46.3 m/s) and theta is the angle of elevation (36.0°).
Vy = 46.3 m/s * sin(36.0°)
Vy ≈ 46.3 m/s * 0.5878
Vy ≈ 27.16 m/s
The initial vertical velocity (Vy) is the velocity at which the ball is moving upwards. Since the ball is initially at the same level as the golfer, the vertical displacement (d) will be the difference in height between the green and the golfer. In this case, it is 5.80 m.
d = 5.80 m
Now, we can use the kinematic equation for vertical motion to find the time of flight (t):
d = Vy * t + (1/2) * g * t^2
Substituting the known values:
5.80 m = 27.16 m/s * t + (1/2) * 9.8 m/s^2 * t^2
Simplifying the equation:
4.9 t^2 + 27.16 t - 5.80 = 0
This is a quadratic equation. We can use the quadratic formula to solve for t:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 4.9, b = 27.16, and c = -5.80.
Solving for t using the quadratic formula:
t = (-27.16 ± √(27.16^2 - 4 * 4.9 * -5.80)) / (2 * 4.9)
t ≈ 1.08 s (ignoring the negative value since time cannot be negative)
Therefore, the ball is in the air for approximately 1.08 seconds before hitting the green.