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)
1: Use the standard equation in x- axis for time taken to travel 15+S meter.
2: It will take same time to travel in the Y direction. The Y distance traveled is 6+S(Tan30)
3: use y(t)=V0(t)-5t^2. Substitute for t from 1 above. Then it is school level maths.
4: Once you get distance S then substitute in 1: for time
Good luck
for y(t) do you use the value in the x or y direction?
or could you show me your full working?
To answer the first question, we need to find the horizontal distance s at which the stone hits the roof.
To find s, we can break down the stone's initial velocity into its horizontal and vertical components. The horizontal component of the initial velocity (v0x) can be found using the equation v0x = v0 * cos(α), where v0 is the initial speed (20 m/s) and α is the angle at which the stone is thrown (35 degrees).
Using the given values, v0x = 20 m/s * cos(35°) ≈ 16.403 m/s.
Next, we can calculate the time it takes for the stone to reach the roof by using the vertical motion equation s = v0y * t + (1/2) * g * t^2, where v0y is the vertical component of the initial velocity, g is the gravitational acceleration (10 m/s^2), and t is the time taken to reach the roof.
The vertical component of the initial velocity (v0y) can be found using the equation v0y = v0 * sin(α), where v0 is the initial speed (20 m/s) and α is the angle at which the stone is thrown (35 degrees).
Using the given values, v0y = 20 m/s * sin(35°) ≈ 11.478 m/s.
At the moment the stone hits the roof, its vertical displacement is equal to the height of the roof (h = 6 m). Therefore, we can set s = h and solve for t.
6 m = 11.478 m/s * t - (1/2) * 10 m/s^2 * t^2.
This is a quadratic equation in t. Simplifying:
5t^2 - 11.478t + 6 = 0.
Solving this quadratic equation using the quadratic formula, we find two possible values for t. However, we discard the negative value since time cannot be negative in this context.
So, the time it takes for the stone to reach the roof is approximately t ≈ 1.632 seconds.
To find the horizontal distance from the house wall where the stone hits the roof (s), we can use the equation s = v0x * t. Plugging in the values:
s = 16.403 m/s * 1.632 s ≈ 26.754 meters.
Therefore, the stone will hit the roof at a horizontal distance of approximately 26.754 meters from the house wall.