31. (II) a shotputter throws the shot with an initial speed of 15.5m/s at a 34.0* angle to the horizontal. Calculate the horizontal distance traveled by the shots if it leaves the athlete’s hand at a height of 2.20m above the ground.

Break the initial speed into vertical and horizontal components.

Determine the time to hit the ground from the vertical velocity
hf=hi+Viv*t-1/2 g t^2
solve for t.

Then, horizonaldistance= vih*t

To calculate the horizontal distance traveled by the shot, we need to break down the initial velocity into its horizontal and vertical components.

1. Calculate the initial horizontal velocity (Vx):
Vx = V * cos(θ)
where V is the initial speed (15.5 m/s) and θ is the launch angle (34.0°).

Vx = 15.5 m/s * cos(34.0°)
Vx ≈ 12.83 m/s

2. Calculate the time of flight (t):
We can use the vertical motion of the shot to find the time it takes to reach the ground.
The vertical displacement (Δy) can be calculated using:
Δy = Vy * t + (1/2) * g * t^2
where Vy is the initial vertical velocity, Δy is the displacement (2.20m), and g is the acceleration due to gravity (9.8 m/s^2).

Since the shot starts and ends at the same height, the displacement in the vertical direction is zero: Δy = 0.
0 = Vy * t + (1/2) * g * t^2
0 = (15.5 m/s * sin(34.0°)) * t + (1/2) * (9.8 m/s^2) * t^2

Solving this quadratic equation will give us the time of flight, t, which we'll need in the next step.

3. Calculate the time of flight (t):
Using the quadratic formula, t can be found using the equation:
t = (-b ± √(b^2 - 4ac)) / (2a)
where a = (1/2) * (9.8 m/s^2), b = (15.5 m/s * sin(34.0°)), and c = 0.

t = [ - (15.5 m/s * sin(34.0°)) ± √((15.5 m/s * sin(34.0°))^2 - 4 * (1/2) * (9.8 m/s^2) * 0)] / [2 * (1/2) * (9.8 m/s^2)]

Solve for t using the positive square root, as negative time does not make physical sense.

4. Calculate the horizontal distance traveled (D):
Since the horizontal velocity remains constant throughout the motion, we can use the formula:
D = Vx * t
where Vx is the initial horizontal velocity (12.83 m/s) and t is the time of flight calculated in the previous step.

D = 12.83 m/s * t

Once you solve the quadratic equation in step 3, substitute the value of t into step 4 to find the horizontal distance traveled by the shot.