A balloon is rising 4 meters per second. If a wind is blowing horizontally at a speed of 2.5 meter per second, find the velocity of the balloon relative to the ground.
That would, of course be
√(4^2+2.5^2)
at an angle θ from the vertical, such that tanθ = 2.5/4
Well, despite their differences in speed, the balloon and the wind seem to be quite cooperative. It's like those odd couples in romantic comedy movies. Now, to find the velocity of the balloon relative to the ground, we just need to add their speeds together.
The balloon is rising at 4 meters per second, while the wind is blowing horizontally at 2.5 meters per second. So, if we add these two lovely speeds, the velocity of the balloon relative to the ground would be 6.5 meters per second. The balloon and the wind have definitely found their groove in this sky-high dance.
To find the velocity of the balloon relative to the ground, we can add the velocity of the balloon relative to the wind to the velocity of the wind.
Given:
- Velocity of the balloon relative to the ground = ?
- Velocity of the balloon relative to the wind = 4 m/s (upward)
- Velocity of the wind = 2.5 m/s (horizontal)
To add the velocities, we need to combine the magnitude and direction of the velocities.
Since the balloon is rising vertically and the wind is blowing horizontally, we need to find the hypotenuse of a right triangle with sides of 4 m/s and 2.5 m/s.
Using the Pythagorean theorem:
Velocity of the balloon relative to the ground = √((velocity of balloon relative to wind)^2 + (velocity of wind)^2)
Velocity of the balloon relative to the ground = √((4 m/s)^2 + (2.5 m/s)^2)
Velocity of the balloon relative to the ground = √(16 m^2/s^2 + 6.25 m^2/s^2)
Velocity of the balloon relative to the ground = √(22.25 m^2/s^2)
Velocity of the balloon relative to the ground ≈ 4.71 m/s
Therefore, the velocity of the balloon relative to the ground is approximately 4.71 meters per second.
To find the velocity of the balloon relative to the ground, we need to combine the vertical velocity due to its rising motion with the horizontal velocity due to the wind.
The vertical velocity of the balloon is given as 4 meters per second, which means it is rising at a speed of 4 meters per second vertically.
The horizontal velocity of the wind is given as 2.5 meters per second, which means it is blowing horizontally at a speed of 2.5 meters per second.
To find the velocity of the balloon relative to the ground, we need to calculate the resultant velocity by combining the vertical and horizontal velocities. This can be done using vector addition.
We can use the Pythagorean theorem to find the magnitude of the resultant velocity. The magnitude of the resultant velocity is given by the square root of the sum of the squares of the vertical and horizontal velocities. Mathematically, it can be written as:
Resultant velocity = √(vertical velocity^2 + horizontal velocity^2)
In this case, the vertical velocity is 4 m/s and the horizontal velocity is 2.5 m/s. Therefore, the magnitude of the resultant velocity is:
Resultant velocity = √(4^2 + 2.5^2) = √(16 + 6.25) = √22.25 = 4.71 m/s
So, the velocity of the balloon relative to the ground is 4.71 meters per second.