Math Problem Statement
You throw a small rock straight up from the edge of a highway bridge that crosses a river. The rock passes you on its way down, 9.00 s after it was thrown. What is the speed of the rock just before it reaches the water 21.0 m below the point where the rock left your hand? Ignore air resistance.
Solution
To solve this problem, we can break it down into two parts:
-
Time for the rock to return to your hand:
- The total time for the rock to go up and come back down past you is given as .
- This means the time for the rock to reach its maximum height and then return to your hand is (half of 9 seconds).
-
Calculating the initial velocity:
-
We'll use the kinematic equation to calculate the initial velocity: where:
- (the velocity at the maximum height is zero),
- is the initial velocity (which we are solving for),
- (the acceleration due to gravity),
- .
Solving for : So, the initial velocity of the rock is .
-
-
Finding the velocity just before it reaches the water:
- Now, we need to determine the velocity of the rock just before it hits the water 21 meters below the starting point.
- We use the kinematic equation:
where:
- (the rock has reversed direction and is falling),
- ,
- (the distance to the water).
Plugging in the values:
Thus, the speed of the rock just before it reaches the water is approximately .
Would you like more details on any of these steps or have any further questions?
Here are 5 related questions to consider:
- How would the result change if we accounted for air resistance?
- What is the maximum height the rock reaches above the bridge?
- What is the total time taken for the rock to reach the water?
- How would the problem change if the rock was thrown downward instead?
- What if the initial velocity was twice as large?
Tip: Always split motion problems into parts—one for each phase of the movement, like upward and downward, to simplify calculations.
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Math Problem Analysis
Mathematical Concepts
Kinematics
Projectile Motion
Free Fall
Formulas
v_f = v_0 - g t
v^2 = v_0^2 + 2 g h
Theorems
Kinematic Equations of Motion
Suitable Grade Level
Grades 10-12
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