A seed shoots out from the pod with the speed of 2.4 m/s but with a direction of motion 30° below the horizontal. The seed pod is initially 1.2 m above the ground.
(a) How long does it take for the seed to land? (Neglect air resistance.)
(b) What horizontal distance does it cover during its flight?
vertical speed initially: 2.4*sin30
time to ground:
h=vi*t+1/2 g t^2
1.2=2.4*sin30*t+4.9t^2
put this in quadratic form, and solve for t using the quadratic equation.
horizonal distance= 2.4cos30*t
To find the answers to both questions, we can break down the motion into horizontal and vertical components. Let's start by finding the time it takes for the seed to land.
(a) How long does it take for the seed to land?
Since the motion is in a projectile path, we can use the equations of motion to calculate the time.
The vertical motion can be described by the equation:
y = y_0 + v_0y * t + (1/2) * a_y * t^2
where:
y = final vertical displacement (0 in this case, as it lands on the ground)
y_0 = initial vertical displacement (1.2 m)
v_0y = initial vertical velocity (v₀ * sinθ, where v₀ = 2.4 m/s and θ = 30°)
a_y = vertical acceleration (in this case, it is the acceleration due to gravity, -9.8 m/s², pointing downward)
t = time taken
Substituting the values into the equation, we get:
0 = 1.2 + (2.4 * sin(30°)) * t + (1/2) * (-9.8) * t^2
Simplifying the equation, we have:
-4.9 * t^2 + 1.2 + 1.2 * t = 0
This is a quadratic equation. Solving it, we find two possible solutions for t. However, since we are looking for the time it takes for the seed to land, we only consider the positive solution.
Using the quadratic formula, we get:
t = (-b + √(b² - 4ac)) / (2a)
Substituting the values from the equation, we have:
t = (-(1.2) + √((1.2)^2 - 4 * (-4.9) * (1.2))) / (2 * (-4.9))
Calculating this expression, we find t ≈ 0.515 s. Therefore, it takes approximately 0.515 seconds for the seed to land.
(b) What horizontal distance does it cover during its flight?
To find the horizontal distance covered, we can use the horizontal motion equation:
x = x_0 + v_0x * t
where:
x = horizontal displacement
x_0 = initial horizontal displacement (0, as the seed is shot horizontally)
v_0x = initial horizontal velocity (v₀ * cosθ, where v₀ = 2.4 m/s and θ = 30°)
t = time taken (0.515 s from part a)
Substituting the values into the equation, we have:
x = 0 + (2.4 * cos(30°)) * 0.515
Calculating this expression, we find x ≈ 1.04 m. Therefore, the seed covers approximately 1.04 meters horizontally during its flight.