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)
The time to rich the highest point is
tₒ= 2vₒ•sinα/2g =2•20•sin35/2•10 =1.15 s
The time for covering the distance to the wall ‚d’ is
t₁=d/vₒ•cosα =15/20•cos35= 0.92 s
At horizontal distance d from the initial point the ball is at the height
h₁=vₒ•sinα•t₁ -gt₁²/2 =
=20•sin25•0.92 -10•0.92²/2 =6.31 m.
The highest position of the ball moving as projectile is
H= vₒ²•sin²α/2g =20²• sin²35/2•10 = 6.38 m.
Therefore, the ball meets the roof at its upward motion, =>
d+s=vₒ•cosα•t …..(1)
h+s•tanβ= vₒ•sinα•t - gt²/2 …..(2)
From (1)
s = vₒ•cosα•t -d
Substitute ‘s’ in (2)
h +tanβ(vₒ•cosα•t –d) =
=vₒ•sinα•t - gt²/2,
6+ 0.58(20•0.82•t -15) = 20•0.57•t- 5t²,
5t² -1.9t-2.7 =0
t=0.95 s.
s= vₒ•cosα•t –d=
=20•0.82•0.95 – 15=
=15.58 – 15 =0.58 m
Answer isn't coming
To find the answer to these questions, we can use the equations of motion to analyze the projectile's trajectory. Let's break it down step by step:
(a) To determine the horizontal distance at which the stone hits the roof, we need to find the projectile's range. The range is the horizontal distance covered by a projectile before it hits the ground or any other surface. In this case, our target surface is the roof of the house.
The range of a projectile can be calculated using the following formula:
Range (R) = (v0^2 * sin(2α)) / g
where:
- v0 is the initial velocity of the projectile,
- α is the angle at which the projectile is launched, and
- g is the acceleration due to gravity.
Using the given values, we can substitute them into the formula:
Range = (20^2 * sin(2 * 35)) / 10
To find the value of sin(2 * 35), we can use the double-angle identity for sine:
sin(2θ) = 2 * sin(θ) * cos(θ)
Using this identity:
Range = (20^2 * 2 * sin(35) * cos(35)) / 10
Now, we can calculate the value of sin(35) and cos(35) using a scientific calculator or by looking them up in a trigonometric table.
Once we have the numerical values of sin(35) and cos(35), we can substitute them back into the equation to calculate the range.
(b) To find the time it takes for the stone to reach the roof, we need to determine the time of flight. The time of flight is the total time taken by the projectile to reach its highest point and then fall back down to the same vertical level.
The time of flight (T) can be calculated using the formula:
T = (2*v0*sin(α)) / g
Substituting the given values into the formula:
T = (2 * 20 * sin(35)) / 10
Now, we can calculate sin(35) using a scientific calculator or by looking it up in a trigonometric table and substitute the value back into the equation to find the time of flight.
By following these steps, we can solve for both the horizontal distance (a) and the time (b) it takes for the stone to hit the roof.