Standing on a balcony, you throw your keys to a friend standing on the ground below. 1.1 seconds after you release the keys, they have an instantaneous velocity of 15.8 m/s, directed 60° below the horizontal. What initial velocity did you give them? magnitude:
direction: below the horizon
Well, the horizontal component is constant.
Horizontal V=15.8*cos60= you do it.
Now in the vertical, vf(1.1)=Viv+gt this assumes the keys were thrown downward, there is another solution..
15.8*sin60=Viv+9.8(1.1)
solve for Viv
then, intial velocity= horizonal in x + verticalinitial in y
magnitude= sqrt(hor^2+viv^2)
angledownward= arctan downward/horizontal
my last two lines show you how to compute. What do you not understand about those lines?
i don't understand what numbers yu plug into this:
magnitude= sqrt(hor^2+viv^2)
angledownward= arctan downward/horizontal
hor= horizontal component found first.
viv= initial vertical velocty, from 15.8*sin60=Viv+9.8(1.1)
arc tan means the angle whose tangent is (vertical velocity initial/horizontal initial)
i don't understand how to get the degree part of this question
To find the initial velocity of the keys, we can use the fact that the instantaneous velocity of the keys 1.1 seconds after being released is 15.8 m/s at an angle of 60° below the horizontal.
The initial velocity can be broken down into horizontal and vertical components. The horizontal component will remain constant throughout the motion, while the vertical component will be affected by gravity.
Let's first find the vertical component of the velocity. Since the keys are moving below the horizon, the vertical component will be negative. We can use the equation:
Vf = Vi + at
Where Vf is the final velocity (15.8 m/s) and Vi is the initial velocity in the vertical direction (which we want to find). Since the keys are falling down, the acceleration due to gravity (a) is -9.8 m/s² (negative because it acts downward). We can assume that the keys are released from rest (Vi = 0), so the equation becomes:
15.8 m/s = 0 + (-9.8 m/s²) * 1.1 seconds
Simplifying, we get:
Vi = 15.8 m/s + 10.78 m/s
Vi = 26.58 m/s (rounded to two decimal places)
Now, let's find the horizontal component of the velocity. Since the keys are thrown below the horizon, the initial horizontal component will be the same as the final horizontal component, because no external forces act in the horizontal direction. Thus, the horizontal component of the initial velocity is equal to the horizontal component of the final velocity.
To find the horizontal component of the final velocity, we can use the equation:
Vfx = Vix
Where Vfx is the final horizontal velocity (which we want to find) and Vix is the initial horizontal velocity. The initial horizontal velocity can be found using trigonometry. We have the magnitude of the initial velocity (Vix), and we know the angle (60°) at which it is below the horizontal. The horizontal component can be found using the equation:
Vix = Vi * cosθ
Where θ is the angle (60°) and Vi is the magnitude of the initial velocity (26.58 m/s). Substituting these values into the equation, we get:
Vix = 26.58 m/s * cos(60°)
Vix = 26.58 m/s * 0.5
Vix = 13.29 m/s (rounded to two decimal places)
So, the initial velocity given to the keys is approximately 13.29 m/s below the horizon.