Question

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.

Answer 1

the snowball's horizontal speed as it starts to fall is

8 cos 40° = 6.13 m/s

Since s = 1/2 at^2, it takes 1.69 seconds to fall 14 meters.

It takes only 1.58s to fall to the height of the man's head.

So the ball moves horizontally 6.13*1.58 = 9.69 meters. Looks like it will sail over the man's head.

Extra credit. How far above the man's head will the snowball be as it flies over?

Answer 2

[AI-generated answer — Explain Bot]

To determine whether the man will be hit by the snowball, we need to analyze the horizontal and vertical motion of the snowball as it rolls off the roof.

First, let's break down the motion into its components:

Vertical motion: The snowball falls vertically downward due to the force of gravity. We can use the equation for vertical displacement:

𝑦 = 𝑣₀𝑡 + ½𝑎𝑡²

Here, 𝑦 represents the vertical displacement, 𝑣₀ is the initial vertical velocity (which is 0 m/s since the snowball is only rolling horizontally off the roof), 𝑎 is the acceleration due to gravity (-9.8 m/s²), and 𝑡 is the time.

Horizontal motion: The snowball rolls horizontally off the roof, so its horizontal displacement is given by:

𝑥 = 𝑣𝑥𝑡

Here, 𝑥 represents the horizontal displacement, 𝑣𝑥 is the horizontal velocity (which is constant and equal to 8 m/s), and 𝑡 is the time.

We need to find the time it takes for the snowball to reach the ground. To compute that, we'll use the vertical motion equation and set 𝑦 (vertical displacement) equal to the height of the roof.

Since the edge of the roof is 14 m above the ground, we have:

14 = 0 + 0.5(-9.8)𝑡²

Simplifying the equation, we get:

0.5(-9.8)𝑡² = 14

Dividing by -4.9:

𝑡² = -14/(-4.9)

𝑡² = 2.8571

Taking the square root of both sides:

𝑡 ≈ 1.69 s

Now that we have the time it takes for the snowball to fall, we can use the horizontal motion equation to find the horizontal displacement. Since the snowball is rolling at a constant horizontal velocity of 8 m/s, we can substitute the values into the equation:

𝑥 = (8)(1.69)

𝑥 ≈ 13.52 m

The horizontal displacement of the snowball is approximately 13.52 m.

Now let's determine the position of the man. Since the man is standing 4 m from the top of the edge of the roof, his horizontal position can be expressed as:

𝑥ₘ = 4 m

Therefore, the horizontal distance between the snowball and the man is given by:

𝑥 - 𝑥ₘ = 13.52 m - 4 m

𝑥 - 𝑥ₘ ≈ 9.52 m

The horizontal distance between the snowball and the man is approximately 9.52 m.

Since the horizontal distance between the snowball and the man is greater than 0, we can conclude that the man will not be hit by the snowball.