A snowball rolls of a roof by accident. The roof is sloped 40.0
o
relative to the horizontal. The edge of
the roof is 14.0 m above the ground, and the speed of the snowball is 7.00 m/s as it rolls off the roof.
Ignore air resistance. Coincidentally, there is a group of management students standing 4.00 m from
the building.
(a) How far from the building does the snowball land?
(b) What is the speed of the snowball at impact?
(c) Did our snowball hit the management students, I mean were the management students ’accidently’
hit by the snowball?
Im just wondering how can you find the speed of the snowball at impact?
initial vertical speed=-7*sin40
initial horizontal speed=7*cos40
time in air:
d=vi*t-1/2 g t^2
d=height, vi=-7sin40, solve for time t.
speed at impact:
vertical: vf=vi+gt
horizontal=7cos40
v=sqrt(vertical^2+horiz^2)
(a) Well, if you take a closer look, you might guess that the snowball will land further away from the building because it has some horizontal velocity due to the sloped roof. To find out exactly how far it lands, we can use some physics magic.
We can break down the initial velocity of the snowball into horizontal and vertical components. Since the speed is 7.00 m/s, we can use trigonometry to find the horizontal component of the velocity. The formula to find this component is:
horizontal velocity = speed * cos(angle)
Using this formula, we find the horizontal velocity to be 7.00 m/s * cos(40.0°).
Now that we have the horizontal velocity, we can use some kinematic equations to determine how long the snowball is in the air before it hits the ground. Since there is no vertical acceleration (we assume no air resistance), we can use the equation:
vertical displacement = initial vertical velocity * time + (1/2) * acceleration * time^2
Since the snowball starts from rest vertically, the initial vertical velocity is 0 m/s. The acceleration due to gravity is -9.8 m/s^2 (negative because it's going down). We want to solve for time, so we can rearrange the equation to:
time = (vertical displacement) / (1/2 * acceleration)
Plugging in the values, we have:
time = (14.0 m) / (1/2 * -9.8 m/s^2)
Finally, we can calculate the horizontal distance traveled using the horizontal velocity and the time:
horizontal distance = horizontal velocity * time
You might want to grab your calculator and do the math.
(b) To find the speed of the snowball at impact, we can use the horizontal and vertical components of the velocity. The vertical velocity will remain constant throughout (assuming no air resistance), so the speed at impact will be the same as the vertical component of the velocity. You can use the equation:
vertical velocity = initial vertical velocity + acceleration * time
Since the initial vertical velocity is 0 m/s and the acceleration due to gravity is -9.8 m/s^2, you can plug in the values and calculate the answer. Again, don't forget your calculator!
(c) As much as I'd like to say that the management students were "accidentally" hit by the snowball, we can determine that by looking at the horizontal distance between the building and the students. If the horizontal distance is less than 4.00 m, then yes, the students were hit. Otherwise, they were spared from the snowball attack.
To find the answers to these questions, we can use the principles of projectile motion. Let's break it down step by step:
(a) To determine how far from the building the snowball lands, we need to analyze the horizontal motion. Since there is no horizontal acceleration, the horizontal velocity remains constant throughout the entire projectile motion. We can use the formula:
Horizontal distance = Horizontal velocity × Time
Since the time of flight is the same for both horizontal and vertical motion, we need to find the time it takes for the snowball to reach the ground. We can use the vertical motion formulas to find the time of flight:
Vertical distance = Initial vertical velocity × Time + (0.5) × Acceleration due to gravity × Time^2
Since the snowball starts from rest in the vertical direction, the equation simplifies to:
Vertical distance = (0.5) × Acceleration due to gravity × Time^2
Substituting the known values:
14.0 m = (0.5) × 9.8 m/s^2 × Time^2
Solving for Time:
Time^2 = (14.0 m) / [(0.5) × 9.8 m/s^2]
Time^2 = 2.857 s^2
Time ≈ 1.69 s
Now, using the time of flight, we can find the horizontal distance:
Horizontal distance = Horizontal velocity × Time
Horizontal distance = 7.00 m/s × 1.69 s
Horizontal distance ≈ 11.83 m
Therefore, the snowball lands approximately 11.83 meters from the building.
(b) To find the speed of the snowball at impact, we need to analyze the vertical motion. The vertical velocity at impact is given by the formula:
Vertical velocity at impact = Initial vertical velocity + (Acceleration due to gravity × Time)
The initial vertical velocity can be found using the formula:
Initial vertical velocity = Vertical velocity × sine(angle)
Substituting the values:
Initial vertical velocity = 7.00 m/s × sine(40.0°)
Initial vertical velocity ≈ 4.517 m/s
Now, substituting all the values into the formula for vertical velocity at impact:
Vertical velocity at impact = 4.517 m/s + (9.8 m/s^2 × 1.69 s)
Vertical velocity at impact ≈ 19.71 m/s
The speed of the snowball at impact is approximately 19.71 m/s.
(c) To determine if the snowball hit the management students, we need to compare the distance of the students from the building (4.00 m) with the horizontal distance we calculated (11.83 m). Since 4.00 m is less than 11.83 m, the snowball landed beyond the group of management students. Therefore, the management students were not hit by the snowball.