a stone is projected vertically upward with a velocity of 20,two seconds later a second stone is similarly projected with the same velocity, when the two stones meet,the second one is rising at a velocity of 10, calculate the length of time the second stone is in motion before they meet
To calculate the length of time the second stone is in motion before they meet, we need to use the equations of motion.
Let's break down the problem and find the necessary information:
For the first stone:
Initial velocity (u1) = 20 m/s (upward)
Time (t1) = 2 seconds
For the second stone:
Initial velocity (u2) = 20 m/s (upward)
Velocity when they meet (v2) = -10 m/s (downward)
Since the stones meet each other, their displacements must be equal. The displacement of an object in motion can be calculated using the following formula:
Displacement (s) = Initial velocity (u) * Time (t) + 0.5 * Acceleration (a) * Time (t)^2
Assuming that the acceleration due to gravity is 9.8 m/s^2, the displacement for each stone can be calculated as follows:
For the first stone:
s1 = u1 * t1 + 0.5 * (-9.8) * t1^2 (Note the negative sign for acceleration, as it opposes the upward motion)
For the second stone:
s2 = u2 * t2 + 0.5 * (-9.8) * t2^2 (Note the negative sign for acceleration, as it opposes the upward motion)
Now, let's equate the displacements, as they meet at the same position:
s1 = s2
By substituting the respective values, the equation becomes:
u1 * t1 + 0.5 * (-9.8) * t1^2 = u2 * t2 + 0.5 * (-9.8) * t2^2
Substituting the given values:
20 * 2 + 0.5 * (-9.8) * 2^2 = 20 * t2 + 0.5 * (-9.8) * t2^2
Simplifying further:
40 - 19.6 = 20 * t2 - 4.9 * t2^2
19.6 = 20 * t2 - 4.9 * t2^2
Rearranging the equation:
4.9 * t2^2 - 20 * t2 + 19.6 = 0
This is a quadratic equation. We can either solve it by factoring or using the quadratic formula. In this case, let's use the quadratic formula:
t2 = (-b ± √(b^2 - 4ac))/(2a)
Substituting the values:
t2 = (-(-20) ± √((-20)^2 - 4 * 4.9 * 19.6))/(2 * 4.9)
Simplifying further:
t2 = (20 ± √(400 - 384.16))/9.8
t2 = (20 ± √(15.84))/9.8
Taking the positive value:
t2 = (20 + √(15.84))/9.8
Calculating the value:
t2 ≈ 1.007 seconds (approximately)
Therefore, the second stone is in motion for approximately 1.007 seconds before they meet.