A golfer, standing on a fairway, hits a shot to a green that is elevated 5.33 m above where she is standing. If the ball leaves her club with a velocity of 48.0 m/s at an angle of 35.7 ° above the ground, find the time that the ball is in the air before it hits the green.
Would I use the equation
y0 + voy t - 1/2gt^2??
h=5.33 m, v(o) = 48 m/s, α = 35.7º
v(oy) = v(o) •sin α = 48• sin35.7º =28 m/s,
The trajectory of the ball is symmetric with respect to the highest point (let it be the point A). Point B (the point where the ball hits the green), point C is at the rising branch of trajectory, and the initial point O.
Now, time OB (which is the question of the problem) =2•( time OA) - time OC. (To imagine make the scheme of trajectory)
OA:
v(y)= v(oy) –gt.
At the point A
v(y) = 0,
0 = v(oy) –g•t.
time OA
t(OA)=v(o) •sin α/g = 28/9.8 =2.86 s.
time OC =t. The height of the point C is h (as well as the point B)
h = v(oy) •t - gt²/2 = 28•t -4.9•t² = 5.33
4.9•t² -28•t + 5.33= 0
t = {28±sqrt(28²-4•4.9•5.33)}/2•4.9 = {28±26}/9.8.
Two roots. We take t= 0.2 because the second root is greater than time OA).
Time OB = 2•2.86 – 0.2 =5.72-0.2=5.7 s.
Yes, you are correct. To find the time that the ball is in the air before it hits the green, you can use the equation for vertical displacement:
y = y0 + voy * t - (1/2) * g * t^2
where:
- y is the vertical displacement (5.33 m, in this case),
- y0 is the initial vertical position (0 m, since the golfer is standing on the fairway),
- voy is the initial vertical velocity (which can be found using the given angle of 35.7° and the initial velocity of 48.0 m/s),
- g is the acceleration due to gravity (approximately 9.8 m/s^2),
- t is the time in seconds.
To find voy using the given angle and initial velocity, you need to decompose the velocity into horizontal and vertical components. The vertical component of the initial velocity (voy) can be found using the equation:
voy = v * sin(theta)
where:
- v is the initial velocity (48.0 m/s, in this case),
- theta is the angle above the ground (35.7°, in this case).
Now you can substitute the values into the equation for vertical displacement and solve for t:
5.33 = 0 + (v * sin(theta)) * t - (1/2) * g * t^2
Simplifying and rearranging the equation, you can solve for t using algebraic methods or numerical methods (such as quadratic formula or graphing calculator).