A projectile is launched at an angle of 64.3 degrees with the horizontal (x axis) at some unknown velocity. After 4 seconds, the object's angle of trajectory is 47.2 degrees above the horizontal. What is the object's vertical displacement at this time in meters?
To find the object's vertical displacement at 4 seconds, we can break down the problem into two components: the vertical and horizontal motion.
First, let's find the initial vertical velocity of the projectile. The initial velocity can be split into horizontal and vertical components using trigonometry.
Vertical component of the initial velocity:
Viy = V * sin(θ), where Viy is the initial vertical component, V is the initial velocity, and θ is the angle of projection.
We only need the vertical component, so we can rewrite the formula for the vertical position as a function of time:
y = Viy * t + (1/2) * g * t^2,
Where y is the vertical displacement, Viy is the vertical component of the initial velocity, t is the time, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
To find the vertical displacement at 4 seconds, we need to determine the value of Viy and substitute it into the formula.
Given:
Angle of projection (θ) = 64.3 degrees
Time (t) = 4 seconds
Angle of trajectory (47.2 degrees above the horizontal)
To find the vertical component of the initial velocity (Viy), we can use the angle of projection:
Viy = V * sin(θ)
Using the sine function:
Viy = V * sin(64.3 degrees)
Now, we need to find the horizontal component of the initial velocity (Vix). We can use the cosine function:
Vix = V * cos(64.3 degrees)
Since the projectile is in motion for 4 seconds and we want to find the vertical displacement at that time, we can calculate the vertical displacement:
y = Viy * t + (1/2) * g * t^2
Substituting the value of Viy and t:
y = (V * sin(64.3 degrees)) * 4 + (1/2) * (9.8 m/s^2) * (4 seconds)^2
Simplifying the equation will give us the vertical displacement in meters.
Note: To solve this problem completely, you will also need to know the initial velocity (V) in order to find the exact vertical displacement.