An object is projected with a velocity of 50 m/s at an angle θ of the vertical of the total time of flight of the projectial is 5sec what is the value of θ?
To find the value of θ, we need to use the given information about the velocity and time of flight of the projectile.
The time of flight of a projectile is the total time it takes for the projectile to reach its highest point and then return to the same vertical level. In this case, the time of flight is given as 5 seconds.
First, let's break down the initial velocity of the projectile into its horizontal and vertical components. The horizontal component of the velocity remains constant throughout the motion, while the vertical component is affected by gravity.
The horizontal component of the velocity (Vx) remains constant, and it can be found using the formula:
Vx = V * cos(θ)
where V is the initial velocity and θ is the angle of projection with respect to the vertical.
In this case, Vx = 50 m/s * cos(θ).
The vertical component of the velocity (Vy) changes due to the force of gravity. At the highest point, the vertical component of the velocity becomes zero (Vy = 0).
To find the time taken to reach the highest point, we can use the formula:
Vy = V * sin(θ) - g * t
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and t is the time taken. Since Vy becomes zero at the highest point, we can substitute Vy = 0 and solve for the time taken to reach the highest point.
0 = V * sin(θ) - g * t
Now we have two equations:
Vx = 50 m/s * cos(θ)
0 = 50 m/s * sin(θ) - 9.8 m/s^2 * t
Since the time of flight is given as 5 seconds, we can substitute t = 5s into the second equation:
0 = 50 m/s * sin(θ) - 9.8 m/s^2 * 5s
Simplifying the equation, we get:
0 = 50 m/s * sin(θ) - 49 m/s^2
Rearranging the terms, we have:
50 m/s * sin(θ) = 49 m/s^2
Finally, we can solve for θ by taking the inverse sine (or arcsine) of both sides:
θ = arcsin(49 m/s^2 / 50 m/s)
Using a scientific calculator or any system with trigonometric functions, we can calculate the value of θ by dividing 49 m/s^2 by 50 m/s and finding the inverse sine of the result:
θ ≈ 53.13 degrees
Therefore, the value of θ is approximately 53.13 degrees.