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 15.4 m/s, directed 60° below the horizontal. What initial velocity did you give them?
15.4@-60=horizontal@0+vertical^-90+g*1.2@-90
ok, then
[email protected]@-90=Horizontal@0-(vertical+1.2g)@-90
setting components equal
15.4sin60=vertical+1.2*9.81 solve for the initial vertical component down.
15.4 cos60=horizontal
now you have the initial horizontal and vertical components, solve for initial velocity
intialvelocity=sqrt(hor^2+vert^2)@arctan vertical/horizontal downward.
To determine the initial velocity you gave the keys, we can use the principles of projectile motion.
First, we need to break down the given information:
- The time it takes for the keys to reach the point with an instantaneous velocity is 1.2 seconds.
- The instantaneous velocity of the keys at that point is 15.4 m/s.
- The direction of the instantaneous velocity is 60° below the horizontal.
To find the initial velocity, we can use the following equation for horizontal motion:
Velocity (horizontal) = Initial velocity × cos(angle)
Since the given velocity is directed 60° below the horizontal, the angle we need to use is the complementary angle, which is 90° - 60° = 30°.
Using the equation, we can rearrange it to solve for the initial velocity:
Initial velocity = Velocity (horizontal) / cos(angle)
Plugging in the values, we have:
Initial velocity = 15.4 m/s / cos(30°)
Using a calculator, we can find the cosine of 30°:
cos(30°) ≈ 0.866
Now, substituting the value, we have:
Initial velocity = 15.4 m/s / 0.866 ≈ 17.79 m/s
Therefore, the initial velocity you gave the keys was approximately 17.79 m/s.