A stone is thrown vertically upward at a speed of 30.30 m/s at time t=0. A second stone is thrown upward with the same speed 2.500 seconds later. At what time are the two stones at the same height?
At what height do the two stones pass each other?
What is the upward speed of the second stone as they pass each other?
To find the time at which the two stones are at the same height, we need to analyze the motion of each stone.
1. Stone 1:
- Initial speed (u1) = 30.30 m/s (thrown upward)
- Time (t1) = 0
2. Stone 2:
- Initial speed (u2) = 30.30 m/s (thrown upward)
- Time (t2) = 2.500 seconds later than Stone 1
To find the time at which the two stones are at the same height, we need to determine the height equation for each stone and set them equal to each other.
The height equation for both stones is given by:
h = ut + (1/2)gt^2
Here, h represents the height, u is the initial speed, t is the time, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
1. Height equation for Stone 1:
h1 = (30.30 * t) - (4.9 * t^2)
2. Height equation for Stone 2:
h2 = (30.30 * (t - 2.500)) - (4.9 * (t - 2.500)^2)
To find the time at which the two stones are at the same height, we set h1 equal to h2 and solve for t:
(30.30 * t) - (4.9 * t^2) = (30.30 * (t - 2.500)) - (4.9 * (t - 2.500)^2)
Simplifying the equation:
30.30t - 4.9t^2 = 30.30t - 75.75 + 4.9t^2 - 24.5t + 30.625
Rearranging the equation:
9.8t^2 - 24.5t - 75.75 + 30.625 = 0
Now we have a quadratic equation, which we can solve to find the value of t when the two stones are at the same height. We can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
In this equation, a = 9.8, b = -24.5, and c = -75.75 + 30.625.
Solving for t using the quadratic formula will give us the time at which the two stones are at the same height.
To find the height at which the two stones pass each other, we can substitute the time value obtained into either height equation (h1 or h2). Let's use h1 for this example:
h = (30.30 * t) - (4.9 * t^2)
Substituting the calculated value of t will give us the height at which the two stones pass each other.
Finally, to find the upward speed of the second stone as they pass each other, we can calculate the velocity of Stone 2 at the time when they pass each other. The velocity equation is the derivative of the height equation, given by:
v = u - gt
Substituting the time value when they pass each other, we can calculate the upward speed of the second stone.