An Alaskan recue plane drops a package of emergency rations to a stranded party of explorers, as illustrated. The plane is traveling horizontally at 100.0m/s at a height of 50.0 m above ground. What horizontal distance does the package tral before striking the ground? I got the answer to that part which was 319. Then it says find the velocity( magnitude and direction) of the package before it hits the ground? I know i have to use cosine and sin but how do i do this??? help please! thanks:)
Your 319 m number is correct.
For the velocity when it hits the ground, use the fact that the horizontal compoinent is 100 m/s and the vertical component is g t, where
t = sqrt (2H/g)
Thus vertical component = sqrt (2 g H)
To find the velocity of the package before it hits the ground, you can use trigonometry. The horizontal and vertical components of the velocity can be calculated using the cosine and sine functions, respectively. Here's how you can approach it:
1. Determine the time it takes for the package to hit the ground:
To find the time, you can use the equation of motion for vertical motion:
h = ut + (1/2)gt^2, where h is the initial height (50.0 m), u is the initial vertical velocity (which is zero as the package is dropped), g is the acceleration due to gravity (-9.8 m/s^2), and t is the time. Rearrange the equation to solve for time (t).
2. Calculate the horizontal component of the velocity:
The horizontal component of the velocity remains constant throughout the motion, so it is equal to the initial horizontal velocity of the plane (100.0 m/s).
3. Calculate the vertical component of the velocity:
The vertical component of the velocity can be found using the equation: v = u + gt, where v is the final vertical velocity, u is the initial vertical velocity (which is zero), g is the acceleration due to gravity (-9.8 m/s^2), and t is the time calculated from step 1.
4. Calculate the magnitude of the velocity:
The magnitude of the velocity can be found using the Pythagorean theorem: velocity^2 = (horizontal velocity)^2 + (vertical velocity)^2. Take the square root of both sides to find the magnitude of the velocity.
5. Calculate the direction of the velocity:
The direction of the velocity can be found by considering the angles involved. The horizontal direction is defined as 0 degrees or east, while the vertical direction is defined as 90 degrees or upward. Use inverse trigonometric functions like arctan to find the angle of the velocity vector.
By following these steps, you should be able to find both the magnitude and direction of the velocity of the package before it hits the ground.