Standing on a balcony, you throw your keys to a friend standing on the ground below. 1.2 seconds after you release the keys, they have an instantaneous velocity of 15.7 m/s, directed 52° below the horizontal. What initial velocity did you give them?
To find the initial velocity, we can break down the motion of the keys into horizontal and vertical components.
First, let's analyze the horizontal component. The keys were thrown with an initial horizontal velocity, and there is no horizontal acceleration (since only the vertical motion is affected by gravity). Hence, the horizontal component of velocity remains constant throughout the motion.
The vertical component of velocity changes due to the effect of gravity. We can use the equation of motion to find the initial vertical velocity. The equation is:
vf = vi + at
Here, vf is the final velocity in the vertical direction, vi is the initial velocity in the vertical direction, a is the acceleration, and t is the time.
In this case, the final velocity (vf) is 15.7 m/s (given), the initial velocity (vi) is what we want to find, the acceleration (a) is the acceleration due to gravity (approximately -9.8 m/s^2, negative because it acts in the opposite direction of motion), and the time (t) is 1.2 seconds.
Using the equation, we can rearrange it to solve for vi:
vi = vf - at
Substituting the values, we have:
vi = 15.7 m/s - (-9.8 m/s^2) * 1.2 s
Simplifying:
vi = 15.7 m/s + 11.76 m/s
The initial vertical velocity is:
vi = 27.46 m/s
Now that we have the initial vertical velocity, we can find the initial velocity magnitude by using the Pythagorean theorem. The initial velocity magnitude is the hypotenuse of a right triangle formed by the horizontal and vertical components of velocity.
Using the equation:
initial velocity magnitude = sqrt((horizontal component)^2 + (vertical component)^2)
We are given that the angle below the horizontal is 52°, so the angle above the horizontal would be 180° - 52° = 128°.
The vertical component of the velocity can be found using the initial vertical velocity:
vertical component = vi * sin(angle)
Plugging in the values:
vertical component = 27.46 m/s * sin(128°)
The horizontal component of the velocity can be found using the initial horizontal velocity:
horizontal component = vi * cos(angle)
Plugging in the values:
horizontal component = 27.46 m/s * cos(128°)
Finally, we can calculate the initial velocity magnitude:
initial velocity magnitude = sqrt((horizontal component)^2 + (vertical component)^2)
Calculating the horizontal and vertical components:
vertical component = 27.46 m/s * sin(128°) ≈ -22.80 m/s
horizontal component = 27.46 m/s * cos(128°) ≈ -12.13 m/s
Calculating the initial velocity magnitude:
initial velocity magnitude = sqrt((-12.13 m/s)^2 + (-22.80 m/s)^2)
initial velocity magnitude ≈ 26.3 m/s
Therefore, the initial velocity given to the keys was approximately 26.3 m/s.