When the dried-up seed pod of a scotch broom plant bursts open, it shoots out a seed with an initial velocity of 2.64 m/s at an angle of 30.0˚ below the horizontal. The seed pod is 0.465 m above the ground.

How long does it take for the seed to land?
What horizontal distance does it cover during its flight?

For the time, solve the equation

y = 0.465 + 2.64 sin30 -(g/2) t^2 = 0
= 0.465 + 1.32 -4.9 t^2
There will be two solutions. Take the positive one.

For the horizonal distance traveled, that would be

X = 2.64 cos30*t = 2.29 t
Use the t from the first part.

shouldn't 2.64 (sin30) by multiplied by time?

y = 0.465 + 2.64 (sin30)t -(g/2) t^2 = 0

Yes, thanks for catching that. :-)

0 = 0.465 + 1.32 t -4.9 t^2
also needed a "t" in it.

Where did you get -4.6 from?

I mean -4.9

To solve this problem, we can use the equations of motion to determine the time it takes for the seed to land and the horizontal distance it covers during its flight.

First, let's consider the vertical motion of the seed. We can use the equation:

h = ut + (1/2)gt^2

where h is the vertical displacement (0.465 m), u is the initial vertical velocity (in this case, it is -2.64 m/s because it is shot below the horizontal), t is the time, and g is the acceleration due to gravity (-9.8 m/s^2).

Plugging in the values, we have:

0.465 = (-2.64)t + (1/2)(-9.8)t^2

Simplifying:

4.9t^2 - 2.64t - 0.465 = 0

Now we can use the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

where a = 4.9, b = -2.64, and c = -0.465.

Solving for t, we get two possible solutions. However, since time cannot be negative, we take the positive value:

t = 0.245 s

So, it takes approximately 0.245 seconds for the seed to land.

To calculate the horizontal distance covered by the seed, we can use the equation:

s = ut

where s is the horizontal displacement, u is the initial horizontal velocity (which is the same as the initial velocity of the seed, 2.64 m/s), and t is the time.

Plugging in the values, we have:

s = 2.64 * 0.245

Simplifying:

s = 0.6456 m

Therefore, the seed covers approximately 0.6456 meters horizontally during its flight.