A rocket is launched into the sky on a windy day. The rocket has a vertical velocity of 15 m/s. There is a strong wind blowing east to west at 35 m/s. How far from the start point is the rocket after 60 seconds?
vertical travel:
y = 15t
horizontal travel:
x = 35t
actual distance:
d = √(x^2+y^2)
= √(15t)^2 + (35t)^2)
= 5√58 t
At t=60, d=2285 m
Assuming the rocket does not run out of fuel and go ballistic (literally)
vertical travel:
y = 15t
horizontal travel:
x = 35t
actual distance:
d = √(x^2+y^2)
= √(15t)^2 + (35t)^2)
= 5√58 t
At t=60, d=2285 m
To find out how far the rocket is from the start point after 60 seconds, we need to consider both the vertical velocity of the rocket and the horizontal velocity of the wind.
First, let's calculate the distance traveled vertically by the rocket after 60 seconds. The rocket has a vertical velocity of 15 m/s, and time is given as 60 seconds. We can use the formula:
distance = velocity × time
So, the vertical distance traveled by the rocket is:
vertical distance = 15 m/s × 60 s
= 900 m
Therefore, the rocket has traveled 900 meters vertically.
Now let's consider the effect of the wind on the horizontal distance. The wind is blowing from east to west at a speed of 35 m/s. In 60 seconds, the wind will have pushed the rocket to the west direction by:
horizontal distance = wind velocity × time
= 35 m/s × 60 s
= 2100 m
So, the rocket has traveled 2100 meters horizontally due to the effect of the wind.
To find the total distance from the start point, we can use the Pythagorean theorem, which states that the square of the hypotenuse (the total distance) is equal to the sum of the squares of the other two sides (vertical and horizontal distances). Mathematically:
total distance² = vertical distance² + horizontal distance²
Substituting the values we calculated earlier, we have:
total distance² = 900² + 2100²
= 810,000 + 4,410,000
= 5,220,000
Taking the square root of both sides, we find:
total distance = √(5,220,000)
≈ 2284.95 m
Therefore, the rocket is approximately 2284.95 meters away from the start point after 60 seconds.