A stone A is thrown vertically upward at 10m/sec. Another stone b is thrown vertically upward at 12m/sec after 1 second at an initial point of B is 2 m higher than the initial point of A.
At what height that two stones have the same elevation?
We can start by using the kinematic equation for the height of a vertically thrown object:
h = vi*t - 0.5*g*t^2
where h is the height, vi is the initial velocity, t is the time, and g is the acceleration due to gravity (9.8 m/s^2).
For stone A, we have:
hA = 10t - 0.5*9.8*t^2
For stone B, we have:
hB = 12(t-1) - 0.5*9.8*(t-1)^2 + 2
Note that we subtract 1 second from the time for stone B since it is thrown after 1 second, and we add 2 meters to account for the initial height difference between A and B.
To find the height at which the two stones have the same elevation, we need to find the value of t that makes hA = hB:
10t - 0.5*9.8*t^2 = 12(t-1) - 0.5*9.8*(t-1)^2 + 2
Simplifying and rearranging, we get:
4.9t^2 - 14t + 8 = 0
Using the quadratic formula, we get:
t = (14 ± sqrt(196 - 4*4.9*8)) / (2*4.9)
t ≈ 2.7 seconds or t ≈ 0.6 seconds
Note that we discard the negative solution since time cannot be negative. Therefore, the two stones have the same elevation at around t = 2.7 seconds.
To find the height at this point, we can plug t = 2.7 seconds into either hA or hB:
hA = 10*2.7 - 0.5*9.8*(2.7)^2 ≈ 37.4 meters
hB = 12(2.7-1) - 0.5*9.8*(2.7-1)^2 + 2 ≈ 37.4 meters
Therefore, the two stones have the same elevation at a height of approximately 37.4 meters.
To find the height at which the two stones have the same elevation, we can use the equations of motion for each stone.
We'll start by considering Stone A:
Using the equation of motion, we have:
v = u + at
Where:
v = final velocity (0 m/s because stone A reaches its highest point)
u = initial velocity (-10 m/s because it is thrown upward)
a = acceleration due to gravity (-9.8 m/s², assuming no air resistance)
t = time taken to reach the highest point
Rearranging the equation, we have:
t = (v - u) / a
Substituting the given values, we get:
t = (0 - (-10)) / (-9.8)
t = 1.02 seconds (approx.)
So, it takes stone A approximately 1.02 seconds to reach its highest point.
Now let's move on to Stone B:
Since Stone B is thrown 1 second after Stone A, the time taken by Stone B to reach the same height will be 1 second less than Stone A, i.e., 1.02 - 1 = 0.02 seconds.
Using the equation v = u + at,
v = (final velocity) = (0 m/s, again reaching its highest point)
u = (initial velocity) = (-12 m/s because it is thrown upward)
a = acceleration due to gravity (-9.8 m/s²)
t = (time) = 0.02 seconds
Again, rearranging the equation and substituting the given values, we have:
0 = -12 - (9.8 * 0.02)
Simplifying further:
0 = -12 + 0.196
12 = 0.196
This equation is not valid, which means that Stone B does not reach the same height as Stone A.
Therefore, there is no specific height at which the two stones have the same elevation. Stone A reaches a higher point than Stone B.
To find the height at which two stones have the same elevation, we need to calculate the time it takes for each stone to reach that height. Let's denote the height as 'h' and the time as 't'.
For stone A, we know that it is thrown vertically upward at a speed of 10 m/s. The initial velocity (u), in this case, is positive (+10 m/s) since it is moving upwards. We can use the following formula of motion to calculate the time it takes to reach height 'h':
h = ut - (1/2)gt^2
Where:
h = height
u = initial velocity
g = acceleration due to gravity (approximately -9.8 m/s^2) since it acts in the opposite direction of the motion
t = time
Plugging in the given values:
h = (10)t - (1/2)(-9.8)t^2
Now, let's work on stone B. We know that it is thrown vertically upward at a speed of 12 m/s, but it starts after 1 second and has an initial height of 2 meters higher than stone A. Therefore, the initial velocity is -12 m/s since it's moving in the opposite direction (downwards), and its initial height (h0) is 2 meters higher than A.
Using the same formula, we can calculate the time it takes for stone B to reach height 'h':
h = h0 + ut - (1/2)gt^2
Plugging in the given values:
h = (2 + 12)t - (1/2)(-9.8)t^2
Now, we need to set both equations equal to each other to find the height at which they intersect:
(10)t - (1/2)(-9.8)t^2 = (2 + 12)t - (1/2)(-9.8)t^2
Simplifying the equation, we get:
(10)t = (2 + 12)t
Now, we can cancel out 't' from both sides of the equation:
10 = 2 + 12
10 = 14
Since we get an inconsistent equation where 10 does not equal 14, it means that the two stones will never have the same elevation.