2.1.6 Another Look at Position, Velocity, and Acceleration

Question 1

Another Look at Position, Velocity, and Acceleration

Answer 1

• The derivative of position with respect to time is velocity. The derivative of velocity with respect to time is acceleration.
• To find the velocity from the acceleration you need to know the velocity at one point in time. The acceleration tells you the slope of the velocity function

Question 2

note

Answer 2

• The top graph is the position function of a ball tossed in the air. The equation for the ball’s path is a parabola with a maximum at t= 2 s: .
• The velocity of the ball is shown in the second graph. Its equation is the derivative with respect to time of the position function: . Initially the velocity of the ball is large and in the positive direction. The velocity is zero at the highest point in the ball’s path (t= 2 s). As the ball falls back to the ground the magnitude of the velocity increases, but the ball moves in the negative direction.
• The ball’s acceleration is shown in the bottom graph. Its equation is the derivative of the velocity function with respect to time: . The acceleration of the ball is constant and it is always negative, or towards the ground.
• An object’s position, velocity, and its acceleration are shown in the top, center, and bottom graph, respectively.
• During the first half of a second, the object moves in the negative direction. Then the object it turns and moves in the positive direction. Its velocity is initially large and negative.At 0.5 s, the object’s velocity is zero. Between 0.5 and 1 s,its velocity is positive and increasing. The acceleration of the object is constant and positive during the entire first second.
• During the time period between 1 and 2 s, the object moves linearly in the positive direction. Because the object’s movement is uniform, its velocity is constant and its acceleration is zero.During the last two seconds the object continues moving in the positive direction, but slows down. Its velocity linearly decreases to nearly zero. The object’s acceleration is constant and negative during this time period.

Question 3

These graphs show the position, velocity, and acceleration of an object. Note that t = 0.6 and t = 3.4 s are indicated on each of the plots. Which of the following statements is correct?

Answer 3

Between 1 and 3.4 s, a is 0, v is constant, and the value of Δx / Δt is constant.

Question 4

Look at the graph of the acceleration of a particle during various stages (A–E). Which of the following correctly describes a possible function for v during the given stage(s)?

Answer 4

In Stage A, the velocity function is −3t.

Question 5

Which of the following statements is not possibly correct?

Answer 5

At a certain time the position function is positive and its slope is a constant negative value. At this same point in time the velocity function is positive and its slope is zero.

Question 6

Which of the following statements is correct?

Answer 6

If the graph of a is a horizontal line and a > 0, then the graph of v is a line with a positive slope and the graph of x is a curve.

Question 7

Suppose that a = 2. Which of the following graphs is not possibly a graph of the velocity function?

Answer 7

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Question 8

Suppose that the position of an object in motion is defined by the equation x = −3t ^2 + 15t + 5. Which of the following is the correct graph for the velocity of this object?

Answer 8

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Question 9

For which of the following functions is a equal to 0?

Answer 9

x = 2t-9

Question 10

Look at the curve of x versus t for a given function. Which of the following functions (of x as a function of t) best fits the given curve?

Answer 10

x = 3t ^2 − 9t

Question 11

For which of the following functions is a a constant but not equal to zero?

Answer 11

x = 3t^ 2 − 9t