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 solve this problem, we can split the motion of the stone into horizontal and vertical components.
For part (a), we need to find the horizontal distance from the house wall at which the stone will hit the roof. We can use the horizontal component of the stone's motion to determine this.
The horizontal component of the initial velocity (v0) can be found using the trigonometric function, cosine.
vx = v0 * cos(α)
Next, we need to find the time it takes for the stone to reach the roof. We can use the vertical component of the stone's motion to determine this.
The vertical component of the initial velocity can be found using the trigonometric function, sine.
vy0 = v0 * sin(α)
To find the time (t) it takes for the stone to reach the roof, we can use the kinematic equation:
h = vy0 * t - (1/2) * g * t^2
Since we know the height of the house wall is 6 meters, we can rearrange the equation to solve for time:
6 = vy0 * t - (1/2) * g * t^2
Once we have the time, we can calculate the horizontal distance (s) by multiplying the horizontal component of the initial velocity by the time:
s = vx * t
Let's calculate both the horizontal distance (s) and the time (t).
Given:
d = 15 m
h = 6 m
β = 30°
v0 = 20 m/s
α = 35°
g = 10 m/s^2
Step 1: Calculate the horizontal component of the initial velocity (vx).
vx = v0 * cos(α)
vx = 20 * cos(35°)
Step 2: Calculate the vertical component of the initial velocity (vy0).
vy0 = v0 * sin(α)
vy0 = 20 * sin(35°)
Step 3: Solve for time (t) by rearranging the kinematic equation.
6 = vy0 * t - (1/2) * g * t^2
Converting β to radians: β = 30° * (π/180) = π/6
6 = (20 * sin(35°)) * t - (1/2) * 10 * t^2
This is a quadratic equation. Let's solve it using the quadratic formula.
Step 4: Calculate the discriminant (D).
D = b^2 - 4ac
a = -(1/2) * g = -(1/2) * 10 = -5
b = (20 * sin(35°)) = approximately 11.36
c = -6
D = (11.36)^2 - 4 * (-5) * (-6)
Step 5: Calculate the time (t) using the quadratic formula.
t = (-b ± √(D)) / (2a)
t = (-(11.36) ± √((11.36)^2 - 4 * (-5) * (-6))) / (2 * (-5))
Solving for t will give us two possible values. We take the positive root since time cannot be negative.
Step 6: Calculate the horizontal distance (s).
s = vx * t
Now you can substitute the values into the equations and calculate the answers for both part (a) and part (b).