Valerie's cast iron piggy bank strikes the ground at a 30 degrees to the horizontal going 250m/sec. assuming level terrain and no air resistance, from how far away did Valerie throw it?
To determine the distance from which Valerie threw the cast iron piggy bank, we need to consider the projectile motion of the object. We can break down the initial velocity into horizontal and vertical components.
Given:
Initial velocity (v₀) = 250 m/s
Angle above horizontal (θ) = 30 degrees
First, we need to find the horizontal component of the initial velocity. We can use trigonometry to do this.
Horizontal component (v₀x) = v₀ * cos(θ)
v₀x = 250 m/s * cos(30°)
v₀x = 250 m/s * (√3/2)
v₀x = 250 m/s * 0.866
v₀x ≈ 216.5 m/s
Now, we can calculate the horizontal distance traveled (range).
Range (R) = (v₀x * time)
Since we're assuming level terrain with no air resistance, the time of flight will be the same for the vertical component as well.
To find the time of flight, we can use the vertical component of the initial velocity.
Vertical component (v₀y) = v₀ * sin(θ)
v₀y = 250 m/s * sin(30°)
v₀y = 250 m/s * (1/2)
v₀y = 250 m/s * 0.5
v₀y = 125 m/s
Since the piggy bank was thrown upward, it will be in the air for twice the time it would take to reach its maximum height. Therefore, we need to calculate the time it takes to reach the maximum height.
Using the equation: v = u + at, where v is final velocity, u is initial velocity, a is acceleration, and t is time:
0 m/s = 125 m/s + (-9.8 m/s²) * t
Now we can solve for time:
125 m/s = 9.8 m/s² * t
t = 125 m/s / 9.8 m/s²
t ≈ 12.76 s
Since the piggy bank reaches its maximum height after 12.76 seconds, it will take twice this time to complete its journey.
Time of flight = 12.76 s * 2
Time of flight ≈ 25.52 s
Finally, we can calculate the range:
Range (R) = v₀x * time
R = 216.5 m/s * 25.52 s
R ≈ 5522.68 m
Therefore, Valerie threw the piggy bank from approximately 5522.68 meters away.