hf=hi+vi*t-4.86t^2
hi=0 hf=-12.7
solve for t.
speed:
vf=vi-9.8t
solve for vf
hi=0 hf=-12.7
solve for t.
speed:
vf=vi-9.8t
solve for vf
Let's break down the problem step by step:
Step 1: Analyze the motion of the stone.
The stone is thrown upward, slows down, reaches the highest point, and then falls downward due to gravitational acceleration. We can consider this as two separate motions: upward motion and downward motion.
Step 2: Calculate the time taken to reach the highest point.
We can use the kinematic equation for vertical motion to determine the time it takes for the stone to reach the highest point:
Δy = v₀y * t + (1/2) * a * t²
Where:
Δy is the vertical displacement (12.7 m)
vâ‚€y is the initial velocity in the vertical direction (6.11 m/s)
a is the acceleration due to gravity (-9.81 m/s²)
t is the time taken to reach the highest point
Plugging in the values, we get:
12.7 m = 6.11 m/s * t + (1/2) * (-9.81 m/s²) * t²
Simplifying the equation, we have:
-4.9 t² + 6.11 t - 12.7 = 0
Now, we can solve this quadratic equation for t using the quadratic formula:
t = (-b ± √(b² - 4ac)) / (2a)
a = -4.9
b = 6.11
c = -12.7
After solving the quadratic equation, we will get two values for t. However, we will only consider the positive value since time cannot be negative in this context.
Step 3: Calculate the time of flight.
The total time the stone remains in the air is twice the time it takes to reach the highest point. So, we multiply the value of t by 2 to determine the total time of flight.
Step 4: Calculate the speed at impact.
To calculate the speed at which the stone impacts the ground, we can use the equation for vertical motion:
v = vâ‚€y + a * t
Where:
v is the final velocity of the stone at impact (to be determined)
vâ‚€y is the initial vertical velocity (6.11 m/s)
a is the acceleration due to gravity (-9.81 m/s²)
t is the time taken to reach the ground
Substituting the values, we get:
v = 6.11 m/s + (-9.81 m/s²) * t
Using the value of t obtained earlier, we can calculate the final velocity at impact.
By following these steps, we can find the speed at which the stone impacts the ground and the time it remains in the air.