Standing on a balcony, you throuw your keys to a friend on the ground below. 1.2 seconds after releasing the keys, they have an instantaneous velocity of 14.8 m/s directed 53 degrees below horizontal. What initial velocity did you give them?
13
To determine the initial velocity you gave the keys, we can break down the given information and use the principles of projectile motion.
Let's assume that the positive x-axis points horizontally in the direction from the balcony to your friend on the ground, and the positive y-axis points vertically upwards. Now let's break down the information given:
- The instantaneous velocity after 1.2 seconds is 14.8 m/s directed 53 degrees below horizontal. This means that the keys have a component of velocity in the horizontal direction (Vx) and a component of velocity in the vertical direction (Vy).
To find the initial velocity (Vo) in terms of its components, we can use the following equations:
Vx = Vo * cos(theta)
Vy = Vo * sin(theta)
Where:
- Vo is the initial velocity magnitude (which we want to find)
- theta is the angle below horizontal (given as 53 degrees)
We can rearrange the equations to solve for Vo:
Vo = Vx / cos(theta)
Vo = Vy / sin(theta)
Now let's calculate the components of velocity:
Vx = 14.8 m/s * cos(53°) ≈ 8.93 m/s
Vy = 14.8 m/s * sin(53°) ≈ 11.56 m/s
Now we can substitute these values into the equations to find Vo:
Vo = 8.93 m/s / cos(53°) ≈ 16.89 m/s
Vo = 11.56 m/s / sin(53°) ≈ 15.17 m/s
Therefore, the initial velocity you gave the keys was approximately 16.89 m/s in the horizontal direction and 15.17 m/s in the vertical direction.