Marie shoots an arrow vertically upward at a sandhill crane. If the arrow travels to a maximum height of 99.5 meters, calculate how fast it was shot.
And, for how much time was Marie's arrow in the air?
To calculate the initial speed of the arrow, we can use the kinematic equation:
v_final = v_initial + (-g) * t,
where:
v_final is the final velocity (which is 0 m/s at the maximum height),
v_initial is the initial velocity (which is what we need to find),
g is the acceleration due to gravity (9.8 m/s^2),
and t is the time it takes for the arrow to reach its maximum height.
Since the final velocity is 0 m/s at the maximum height, we can solve the equation for v_initial:
0 = v_initial + (-9.8) * t.
Given that the maximum height is 99.5 meters, we know that the arrow will take the same amount of time to reach its maximum height and then fall back to the ground. Therefore, the total time the arrow is in the air is twice the time it takes to reach the maximum height.
So, let's solve for t:
99.5 = v_initial * t + (1/2) * (-9.8) * t^2.
Simplifying the equation, we have:
4.9 * t^2 + v_initial * t - 99.5 = 0.
Now, we can use the quadratic formula to solve for t:
t = (-v_initial ± √(v_initial^2 - 4 * 4.9 * (-99.5))) / (2 * 4.9).
Since we are interested in the positive time, we take the positive square root in the formula.
Now, let's solve for the initial velocity.
1. Substitute the values into the formula for t:
99.5 = v_initial * t + (1/2) * (-9.8) * t^2.
2. Rearrange the equation to make it quadratic:
4.9 * t^2 + v_initial * t - 99.5 = 0.
3. Apply the quadratic formula:
t = (-v_initial ± √(v_initial^2 - 4 * 4.9 * (-99.5))) / (2 * 4.9).
4. Solve for t using the positive square root.
5. Now, we can calculate the initial velocity:
v_initial = (-99.5 - (1/2) * (-9.8) * t^2) / t.
Once we have the value for v_initial, we will be able to determine the initial speed at which Marie shot the arrow.
To find the amount of time the arrow was in the air, we know that it takes twice the time to reach the maximum height. So, once we calculate the value of t using the quadratic formula, the total time in the air is 2 * t.