A snowball off a roof that slopes downward at an angle of 40degrees. The edge of the roof is 14 m above the ground and the snowball has a speed of 8 m/s as it rolls off the edge of the roof. If a 1.8 m tall man is standing 4 m from the top of the edge of the roof, will he be hit by the snowball? Give details.
To determine if the man will be hit by the snowball, we need to analyze the path of the snowball after it rolls off the edge of the roof.
First, let's break down the initial velocity of the snowball into horizontal and vertical components. The vertical component of the velocity can be found using the speed and the angle of the roof.
Vertical velocity (Vy) = speed * sin(angle)
Vy = 8 m/s * sin(40°)
Vy ≈ 5.135 m/s
The horizontal component of the velocity remains constant throughout the motion of the snowball. This component is given by:
Horizontal velocity (Vx) = speed * cos(angle)
Vx = 8 m/s * cos(40°)
Vx ≈ 6.112 m/s
Now, let's examine the time it takes for the snowball to reach the ground. We can use the formula for vertical displacement to do this.
Vertical displacement (Sy) = initial vertical velocity * time + 0.5 * acceleration * time^2
Since the snowball is in free fall, the acceleration due to gravity (g) is approximately 9.8 m/s^2.
The vertical displacement from the edge of the roof to the ground is 14 m, so we can set up the equation:
-14 m = 5.135 m/s * t + 0.5 * (-9.8 m/s^2) * t^2
Rearranging the equation, we get a quadratic equation:
0.5 * (-9.8 m/s^2) * t^2 + 5.135 m/s * t - 14 m = 0
Solving this quadratic equation will give us the time it takes for the snowball to reach the ground.
Using the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a)
a = 0.5 * (-9.8 m/s^2)
b = 5.135 m/s
c = -14 m
Calculating the discriminant, b^2 - 4ac:
(b^2 - 4ac) = (5.135 m/s)^2 - 4 * 0.5 * (-9.8 m/s^2) * (-14 m)
(b^2 - 4ac) ≈ 39.65 m^2/s^2
Since the discriminant is positive, there are two real solutions for t. We want to find the positive time it takes for the snowball to reach the ground.
Using the quadratic formula:
t = (-5.135 m/s ± √(39.65 m^2/s^2)) / (2 * 0.5 * (-9.8 m/s^2))
Calculating the time it takes for the snowball to reach the ground:
t ≈ 0.933 s
Now that we know the time it takes for the snowball to fall, let's determine the horizontal distance it travels during this time:
Distance (D) = horizontal velocity * time
D = 6.112 m/s * 0.933 s
D ≈ 5.7 m
Since the man is standing 4 m away from the edge of the roof, and the snowball travels around 5.7 m horizontally, the man will be hit by the snowball.