A stone is thrown upward with a speed of 20 m/s. It is caught on its way down at a point 5.0 m above where it is thrown.
A) How fast was it going when it was caught?
B) how long did the trip take?
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To find the answers to these questions, we need to use the equations of motion in physics. The equations we can use in this case are equations for vertical motion:
1) v = u + at
2) s = ut + (1/2)at^2
3) v^2 = u^2 + 2as
Where:
- v is the final velocity,
- u is the initial velocity,
- a is the acceleration,
- s is the displacement, and
- t is the time.
Let's break down the problem step by step:
A) To find the velocity when the stone was caught, we need to find the final velocity (v) when it reaches a height of 5.0 m above the initial position of the throw. The initial velocity (u) is given as 20 m/s.
Step 1: Determine the displacement (s):
The stone's displacement is given as 5.0 m, which means it traveled upward.
Step 2: Calculate the acceleration (a):
The acceleration due to gravity is acting downwards and is typically denoted as -9.8 m/s^2 (negative because it acts opposite to the direction of motion).
Step 3: Use the equations:
Using equation 2 (s = ut + (1/2)at^2), we can rearrange it to solve for time (t):
5.0 m = (20 m/s)t + (1/2)(-9.8 m/s^2)t^2
0 = -4.9t^2 + 20t - 5.0
We can solve this quadratic equation to find the time it takes for the stone to reach a height of 5.0 m.
B) To find the total time taken for the trip, we need to calculate the time it takes for the stone to reach the maximum height (before falling back down) and double it.
Step 4: Finding the time of flight:
The stone is thrown upwards, and it reaches the maximum height when its final velocity becomes zero (v = 0 m/s). Using equation 1 (v = u + at), we have:
0 = 20 m/s + (-9.8 m/s^2)t
Solve for t, the time it takes for the stone to reach the maximum height, considering the initial velocity and the acceleration due to gravity.
Step 5: Calculate the total time taken:
Multiply the time of flight by 2 to account for the upward and downward motion of the stone.
By following these steps, you should be able to calculate both the velocity when the stone was caught and the total time taken for the trip.
v= -17 m/s
t= 3.81