A particle starts from the origin at t = 0 and moves along the positive x axis. A graph of the velocity of the particle as a function of the time is shown in the figure; the v-axis scale is set by vs = 7.0 m/s. (a) What is the coordinate of the particle at t = 5.0 s? (b) What is the velocity of the particle at t = 5.0 s? (c) What is the acceleration of the particle at t = 5.0 s? (d) What is the average velocity of the particle between t = 1.0 s and t = 5.0 s? (e) What is the average acceleration of the particle between t = 1.0 s and t = 5.0 s?
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To answer these questions, we need to analyze the graph of the velocity as a function of time.
Let's go step-by-step to find the answers:
(a) What is the coordinate of the particle at t = 5.0 s?
To find the coordinate at a given time, we need to integrate the velocity function. Since the particle is moving along the positive x-axis, we can find the displacement using the area under the velocity-time graph.
From the graph, we can see that the velocity is constant at 7.0 m/s from t = 0 s to t = 4 s, and then it becomes 0 m/s from t = 4 s to t = 5 s.
The displacement is given by the area of the rectangle formed by the constant velocity section:
Displacement = Velocity * Time = 7.0 m/s * (4 s - 0 s) = 28 m
Therefore, the coordinate of the particle at t = 5.0 s is 28 m.
(b) What is the velocity of the particle at t = 5.0 s?
The velocity at t = 5.0 s can be read directly from the graph, and it is 0 m/s.
(c) What is the acceleration of the particle at t = 5.0 s?
Since the particle's velocity is constant at 0 m/s from t = 4 s to t = 5 s, the acceleration during this period is zero.
(d) What is the average velocity of the particle between t = 1.0 s and t = 5.0 s?
The average velocity is given by the total displacement divided by the total time.
From t = 1.0 s to t = 4.0 s, the velocity is constant at 7 m/s. So, the total displacement during this period is:
Displacement = Velocity * Time = 7.0 m/s * (4 s - 1 s) = 21 m
From t = 4.0 s to t = 5.0 s, the velocity is 0 m/s, so there is no displacement.
Therefore, the total displacement from t = 1 s to t = 5 s is 21 m. And the total time is 5 s - 1 s = 4 s.
Average Velocity = Total Displacement / Total Time = 21 m / 4 s = 5.25 m/s
Therefore, the average velocity of the particle between t = 1.0 s and t = 5.0 s is 5.25 m/s.
(e) What is the average acceleration of the particle between t = 1.0 s and t = 5.0 s?
Since the velocity is constant during the entire interval from t = 1 s to t = 5 s, the average acceleration is 0 m/s^2.
To determine the coordinate, velocity, acceleration, average velocity, and average acceleration of the particle at t = 5.0 s, we need to analyze the graph of the velocity as a function of time.
(a) The coordinate of the particle at t = 5.0 s can be found by calculating the area under the velocity-time graph from t = 0 to t = 5.0 s. Since the particle is moving along the positive x-axis, the area under the graph represents the displacement (change in position) of the particle. In this case, since the velocity is positive for the entire duration, the displacement is equal to the area under the graph.
To find the area, we need to determine the shape of the graph and calculate the area accordingly. Since the graph is a straight line parallel to the time axis, it represents constant velocity. Therefore, the area is given by the formula:
Area = Velocity * Time
From the graph, we can estimate that the velocity at 5.0 s is approximately 7.0 m/s. Substituting the values, we get:
Area = 7.0 m/s * 5.0 s = 35.0 m
Thus, the coordinate of the particle at t = 5.0 s is 35.0 m.
(b) The velocity of the particle at t = 5.0 s is given directly by the graph. From the graph, we can read off that the velocity is approximately 7.0 m/s.
Therefore, the velocity of the particle at t = 5.0 s is 7.0 m/s.
(c) The acceleration of the particle at t = 5.0 s can be determined by analyzing the slope of the velocity-time graph. Acceleration is the rate of change of velocity with respect to time. On a velocity-time graph, the slope represents the acceleration.
Since the graph is a straight line parallel to the time axis, the slope is zero. This indicates that the acceleration at t = 5.0 s is zero.
(d) The average velocity of the particle between t = 1.0 s and t = 5.0 s can be calculated by finding the total displacement and dividing it by the time interval.
To find the total displacement, we again use the area under the graph. Since the graph is a straight line parallel to the time axis, the area is given by:
Area = Velocity * Time
Substituting the values from the graph, we get:
Area = 7.0 m/s * (5.0 s - 1.0 s) = 28.0 m
The time interval is 5.0 s - 1.0 s = 4.0 s.
Average Velocity = Total Displacement / Time Interval
Average Velocity = 28.0 m / 4.0 s = 7.0 m/s
Therefore, the average velocity of the particle between t = 1.0 s and t = 5.0 s is 7.0 m/s.
(e) The average acceleration of the particle between t = 1.0 s and t = 5.0 s can be calculated by finding the change in velocity and dividing it by the time interval.
To find the change in velocity, we subtract the initial velocity at t = 1.0 s from the final velocity at t = 5.0 s. From the graph, we read that the initial velocity is 0 m/s and the final velocity is 7.0 m/s.
Change in Velocity = Final Velocity - Initial Velocity
Change in Velocity = 7.0 m/s - 0 m/s = 7.0 m/s
The time interval is 5.0 s - 1.0 s = 4.0 s.
Average Acceleration = Change in Velocity / Time Interval
Average Acceleration = 7.0 m/s / 4.0 s = 1.75 m/s²
Therefore, the average acceleration of the particle between t = 1.0 s and t = 5.0 s is 1.75 m/s².