A man chased by the army, leaps with speed 12 m/s at a 47 degree angle above the horizontal from the top of a building. If he lands 40.0 m from the base of the building, then finds the height of the building
Vo = 12m/s[47].
Xo = 12*Cos47 = 8.18 m/s.
Yo = 12*sin47 = 8.78 m/s.
Xo*T = 40.
8.18*T = 40, T = 4.89 s. = Time in air.
Y = Yo + g*Tr_.
0 = 8.78 - 9.8Tr, Tr = 0.896 s. = Rise time.
Tr+Tf = 4.89.
0.896 + Tf = 4.89, Tf = 3.99 s. = Fall time.
ha = Yo*Tr + 0.5g*Tr^2.
ha = 8.78*0.896 - 4.9*0.896^2 = 3.93 m. = Ht. above the roof.
h = 0.5g*Tf^2 = 4.9*3.99^2 = 78 m. Above gnd.
ha+hb = 78 m.
3.93 + hb = 78, hb = 74.1 m. = Ht. of the bldg.
To find the height of the building, we can use the equations of motion and consider the vertical component of the man's leap.
Let's break down the information given:
- The man's initial speed is 12 m/s.
- The angle above the horizontal is 47 degrees.
- The horizontal distance traveled (or range) is 40.0 m.
First, we need to find the time it takes for the man to land. Since we're only concerned with the vertical component of his motion, we can treat it as a projectile motion problem.
The equation that relates horizontal distance, initial velocity, and time is:
Range = (Initial velocity * cosθ) * time
where θ is the angle of projection.
Plugging in the values we have:
40.0 m = (12 m/s * cos 47°) * time
To solve for time, rearrange the equation:
time = 40.0 m / (12 m/s * cos 47°)
Now we can find the time it takes for the man to land.
Next, we can use the equation that relates vertical displacement, initial vertical velocity, time, and acceleration due to gravity:
Vertical displacement = (Initial vertical velocity * time) + (0.5 * acceleration due to gravity * time^2)
Since the man leaps upwards, the vertical displacement will be equal to the height of the building.
The initial vertical velocity can be calculated as:
Initial vertical velocity = Initial velocity * sinθ
Substituting the known values:
Initial vertical velocity = 12 m/s * sin 47°
Now, plug in all the values into the equation for vertical displacement to find the height of the building.