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= 45 ∘. 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 answer these questions, we need to analyze the motion of the stone and calculate the horizontal distance it travels and the time it takes to reach the roof.
(a) Horizontal Distance (s):
To find the horizontal distance from the house wall where the stone hits the roof, we can divide the motion into horizontal and vertical components.
- Vertical Component:
First, we need to find the time it takes for the stone to reach the maximum height. We can use the vertical component of the initial velocity (v0) and the gravitational acceleration (g).
Using the equation:
(vf)² = (vi)² + 2ad
Where:
- vf = final vertical velocity (0 m/s at maximum height)
- vi = initial vertical velocity (v0 * sinα)
- a = acceleration (-g)
- d = vertical displacement (h)
Substituting the known values:
0 = (v0 * sinα)² + 2(-g)h
Simplifying and solving for v0 * sinα:
(v0 * sinα)² = -2(-g)h
v0 * sinα = sqrt(2gh)
Now, we can find the time (t) it takes for the stone to reach the maximum height by using the equation:
vf = vi + at
0 = v0 * sinα - gt
Solving for t:
t = (v0 * sinα) / g
- Horizontal Component:
Next, we need to find the horizontal distance covered by the stone in the time it takes to reach the maximum height.
Using the equation:
d = v * t
Where:
- d = horizontal distance
- v = horizontal component of velocity (v0 * cosα)
- t = time
Substituting the known values:
d = (v0 * cosα) * t
- Final Calculation:
Now that we have the values of t and d, we can calculate the horizontal distance from the house wall where the stone hits the roof.
By substituting v0 = 20 m/s, α = 45°, and t = (v0 * sinα) / g into the equation above:
d = (20 * cos45°) * ((20 * sin45°) / 10)
d = (20 * sqrt(2)/2) * (2/2)
d = 20 * 1 * 1
d = 20 meters
Therefore, the stone is going to hit the roof at a horizontal distance of 20 meters from the house wall.
(b) Time to Reach Roof:
To find the time it takes for the stone to reach the roof, we need to find the total time of flight.
Using the equation:
t_total = 2 * t
Where:
- t_total = total time of flight
- t = time to reach maximum height
Substituting the known values:
t_total = 2 * ((20 * sin45°) / 10)
t_total = 2 * (20/10)
t_total = 2 * 2
t_total = 4 seconds
Therefore, it takes 4 seconds for the stone to reach the roof.