We are standing at a distance d=15 m away from a house. The house wall is h=6 m high and the roof has an inclination angle β=30 ∘. We throw a stone with initial speed v0=20 m/s at an angle α= 35 ∘. The gravitational acceleration is g=10 m/s2. (See figure)
(a) At what horizontal distance from the house wall is the stone going to hit the roof - s in the figure-? (in meters)
(b) What time does it take the stone to reach the roof? (in seconds)
To find the horizontal distance from the house wall where the stone hits the roof (s), we can use the equations of motion for projectile motion. We'll break the initial velocity into horizontal and vertical components.
Step 1: Calculate the initial velocity components:
The horizontal component of the initial velocity (v0x) can be calculated as:
v0x = v0 * cos(α)
The vertical component of the initial velocity (v0y) can be calculated as:
v0y = v0 * sin(α)
Substituting the given values:
v0x = 20 * cos(35 ∘)
v0x ≈ 16.35 m/s
v0y = 20 * sin(35 ∘)
v0y ≈ 11.51 m/s
Step 2: Calculate the time taken to reach the roof (t):
We can use the equation of motion for vertical motion:
h = v0y * t - (1/2) * g * t^2
Substituting the given values:
6 = 11.51 * t - 0.5 * 10 * t^2
Simplifying the equation:
5t^2 - 11.51t + 6 = 0
Using the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
where a = 5, b = -11.51, and c = 6
Calculating the values under the square root:
b^2 - 4ac = (-11.51)^2 - 4 * 5 * 6
b^2 - 4ac ≈ 132.4401 - 120
b^2 - 4ac ≈ 12.4401
Calculating the values of t:
t = (11.51 ± sqrt(12.4401)) / (2 * 5)
t ≈ (11.51 ± 3.52) / 10
Taking the positive root, because we are interested in the stone reaching the roof:
t ≈ (11.51 + 3.52) / 10
t ≈ 1.55 seconds
Step 3: Calculate the horizontal distance (s):
The horizontal distance can be calculated using the equation:
s = v0x * t
Substituting the given values:
s = 16.35 * 1.55
s ≈ 25.33 meters
Answer:
(a) The stone is going to hit the roof at a horizontal distance of approximately 25.33 meters from the house wall.
(b) It takes approximately 1.55 seconds for the stone to reach the roof.