A rocket is launched from a tower that is 200 feet high at 1000ft/sec at 48°.
How far does the rocket travel?
recall that the equation of motion is
y = h0 + tanθ x - g/(2v^2) sec^2θ x^2
so,
y = 200 + 1.11x - 0.00001787x^2
y=0 at x=62278.7 ft or 11.8 miles
To find out how far the rocket travels, we can break down the given information and use trigonometry to calculate the horizontal distance covered by the rocket.
First, let's consider the initial speed of the rocket, which is given as 1000 ft/sec. This speed can be considered as the initial velocity of the rocket in the vertical direction (upwards).
Next, we need to find the horizontal component of this initial velocity. To do that, we can use the given angle of 48°. The horizontal component of the velocity can be found by multiplying the initial velocity (1000 ft/sec) by the cosine of the angle:
Horizontal component of velocity = 1000 ft/sec * cos(48°)
Now, let's calculate the horizontal distance traveled by the rocket using the following formula:
Distance = Horizontal velocity * Time taken
Since we are given the speed in units of ft/sec, the time taken is in seconds. We need to determine the time taken for the rocket to reach the ground.
To calculate the time, we need to consider the vertical component of the motion. The rocket is initially at a height of 200 feet and is subject to acceleration due to gravity (approximately 32.2 ft/sec^2). We can use the equations of motion to calculate the time it takes for the rocket to reach the ground.
Using the equation: Y = Y0 + V0y * t + (1/2) * a * t^2
Where:
- Y is the final position (0 ft, as the rocket reaches the ground)
- Y0 is the initial position (200 ft)
- V0y is the initial vertical velocity component (1000 ft/sec * sin(48°))
- t is the time taken
- a is the acceleration due to gravity (-32.2 ft/sec^2)
Rearranging the equation, we get: t = (sqrt(V0y^2 - 2*a*Y0) - V0y) / a
Now, substitute the known values into the equation and solve for t:
t = (sqrt((1000 * sin(48°))^2 - 2 * -32.2 * 200) - (1000 * sin(48°))) / -32.2
Once you have determined the time taken by the rocket to reach the ground, substitute it back into the formula to calculate the horizontal distance traveled:
Distance = (Horizontal component of velocity) * (Time taken)
Therefore, the distance traveled by the rocket can be obtained by substituting the appropriate values into the formula and performing the calculations.