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?
I got 6.91 meters for part a. Im not sure if that is right or not. Im not sure how to do part b and c.
when it leaves the roof
vertical velocity ... -7.00 sin(40º)
horizontal velocity ... 7.00 cos(40º)
the horizontal velocity is constant
the vertical velocity is increased by gravity (acceleration)
h = -1/2 g t² + v₀ t + h₀ ... t is the time of flight
0 = -4.9 t² - 7.00 sin(40º) t + 14.0
... use the quadratic formula to find t
(a) 7.00 cos(40º) t
(b) use t to find the final vertical velocity
... -7.00 sin(40º) t - g t
... add the horizontal and vertical components of velocity
... use Pythagoras (a² + b² = c²)
(c) only if they are REALLY tall
To solve this problem, we need to analyze the motion of the snowball as it rolls off the roof and reaches the ground. We can use the principles of projectile motion to find its distance from the building (horizontal displacement), its speed at impact, and determine whether it hits the group of management students.
Let's break down the problem into different parts:
(a) How far from the building does the snowball land?
To find the horizontal displacement, we can use the equation:
horizontal displacement = initial horizontal velocity * time of flight.
First, we need to find the time of flight. Since there is no vertical acceleration, we can use the equation:
vertical displacement = (initial vertical velocity * time) + (0.5 * acceleration * time^2),
where the initial vertical velocity is 0 m/s. The vertical displacement is equal to the height of the roof (14.0 m).
14.0 m = 0.5 * 9.8 m/s^2 * time^2
Now, we can solve this equation to find the time of flight (time). Once we have the time of flight, we can calculate the horizontal displacement using the given initial horizontal velocity (7.00 m/s).
(b) What is the speed of the snowball at impact?
To find the speed of the snowball at impact, we can use the principles of conservation of energy. Since there is no air resistance, we can assume that the total mechanical energy (kinetic + potential) of the snowball is conserved.
At the initial position (on the roof), the mechanical energy is:
initial mechanical energy = potential energy = mass * g * height of the roof,
where g is the acceleration due to gravity (9.8 m/s^2).
At the final position (at impact with the ground), the mechanical energy is:
final mechanical energy = kinetic energy + potential energy = (0.5 * mass * v^2) + (mass * g * distance fallen),
where v is the velocity at impact (which we need to find) and the distance fallen is the height of the roof (14.0 m).
By equating the initial and final mechanical energies, we can solve for the velocity at impact (v).
(c) Did our snowball hit the management students?
To determine if the snowball hit the management students, we need to compare the horizontal distance from the building (which we calculated in part a) with the distance of the management students from the building (given as 4.00 m). If the horizontal distance is less than or equal to the distance of the management students, then the snowball hit them.
Now, we can work through the calculations to find the answers to the questions.