A fireworks rocket consists of two fuel stages which serve to provide two consecutive accelerations of the rocket. Starting from rest at a height of 0 m and a time of 0 seconds, the rocket is ignited and the accelerations begin. The first acceleration stage lasts for 1.64 seconds and accelerates the fireworks to an upward velocity of 8.52 m/s. The second acceleration stage lasts for 3.28 seconds and accelerates the fireworks to an upward velocity of 15.34 m/s. The graph at the right depicts the motion. Use the graph to answer the following questions.
(a) Determine the acceleration of the rocket during the first fuel stage.
(b) Determine the acceleration of the fireworks during the second fuel stage (from t1 to t2).
(c) Determine the distance the rocket travels upward (i.e., height above the ground) during the first stage of motion.
(d) Determine the distance the rocket travels upward during the second stage of motion (from t1 to t2).
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To answer these questions, we can use the position-time graph provided. The slope of the graph at any point represents the velocity of the object at that point, and the second derivative of the position function gives us the acceleration.
(a) To determine the acceleration of the rocket during the first fuel stage, we need to find the slope of the graph during that time interval. From the graph, we see that the first acceleration stage lasts for 1.64 seconds, and the rocket reaches an upward velocity of 8.52 m/s. Therefore, the acceleration can be calculated using the following formula:
acceleration = change in velocity / time interval
acceleration = (final velocity - initial velocity) / time interval
acceleration = (8.52 m/s - 0 m/s) / 1.64 s
acceleration ≈ 5.20 m/s²
Therefore, the acceleration of the rocket during the first fuel stage is approximately 5.20 m/s².
(b) To determine the acceleration of the fireworks during the second fuel stage (from t1 to t2), we can again use the slope of the graph during that time interval. From the graph, we see that the second acceleration stage lasts for 3.28 seconds, and the rocket reaches an upward velocity of 15.34 m/s. Using the same formula as above:
acceleration = change in velocity / time interval
acceleration = (final velocity - initial velocity) / time interval
acceleration = (15.34 m/s - 8.52 m/s) / 3.28 s
acceleration ≈ 2.07 m/s²
Therefore, the acceleration of the fireworks during the second fuel stage is approximately 2.07 m/s².
(c) To determine the distance the rocket travels upward during the first stage of motion, we need to find the area under the graph corresponding to the first acceleration stage. Since the graph represents velocity as a function of time, the area under the graph represents the displacement or distance traveled. In this case, the area of interest is the triangular region formed by the line and the x-axis during the first acceleration stage.
To find the area of a triangle, we can use the formula:
area = (base * height) / 2
In this case, the base of the triangle is 1.64 seconds (the time interval of the first acceleration stage) and the height is 8.52 m/s (the final velocity reached in that stage). Plugging these values into the formula:
area = (1.64 s * 8.52 m/s) / 2
area ≈ 7.10 m
Therefore, the distance the rocket travels upward during the first stage of motion is approximately 7.10 meters.
(d) Similarly, to determine the distance the rocket travels upward during the second stage of motion (from t1 to t2), we need to find the area under the graph corresponding to the second acceleration stage. In this case, the area is again the triangular region formed by the line and the x-axis during the second acceleration stage.
Using the same formula as above, the base of the triangle is 3.28 seconds (the time interval of the second acceleration stage) and the height is 15.34 m/s (the final velocity reached in that stage).
area = (3.28 s * 15.34 m/s) / 2
area ≈ 25.27 m
Therefore, the distance the rocket travels upward during the second stage of motion is approximately 25.27 meters.