2.1.3 Understanding Instantaneous Velocity

Question 1

Understanding Instantaneous Velocity

Answer 1

• The instantaneous velocity is the velocity of an object at an instant in time: Dx/Dt = dx/dt .
• The instantaneous velocity is the slope of the tangent to the position function.

Question 2

note

Answer 2

• A ball falls to the ground with nonuniform average velocity as shown by the parabolic position function. The shallow slope of the lower secant indicates that the ball falls slowly during the first half of the time interval, and the steep slope of the upper secant indicates that the ball falls more quickly during the second half.
• To determine the speed of the ball at a specific point in time rather than over an interval, you need to calculate the instantaneous velocity instead of the average velocity.
• The longest secant represents the average velocity of the ball over a 3 s interval. The slopes of the shorter secant lines represent the average velocity of the ball over time intervals of 2 s and 1 s. The line shown on the right side of the image represents the velocity of the ball at t= 1 s as the time interval approaches zero. It is the instantaneous velocity:. The instantaneous velocity has the same slope as the tangent to the position curve at t= 1 s.
• The curve on the left represents a complicated position function of an object.
• At t1, the slope of the tangent is steep and positive so the instantaneous velocity is large and positive.
• At t2, the slope of the tangent and the instantaneous velocity are zero. The object is not moving.
• At t3, the slope of the tangent and the instantaneous velocity are negative. The object moves in the negative x-direction

Question 3

Look at the position function of a falling object. Which of the following statements correctly compares the instantaneous velocities at different points on the plot?

Answer 3

The slope of the tangent line at t1 is less than the slope of the tangent line at t2, which is less than the slope of the tangent line at t3, etc.

Question 4

Instantaneous velocity

Answer 4

Instantaneous velocity is the limit of the ratio of the displacement to the change in time as Δt approaches 0.

Question 5

The displacement of a particle is shown in the plot. Which of the following statements concerning this plot is not correct?

Answer 5

The instantaneous velocities at t3 and t5 are significantly different.

Question 6

Which of the following best defines instantaneous velocity?

Answer 6

the velocity of an object at a particular instant

Question 7

In a demonstration a ball falls directly to the ground. Which of the following statements about this event is not correct? (Assume downward is the positive direction.)

Answer 7

The average velocity during the first half of the fall is greater than the average velocity during the entire experiment.

Question 8

Which of the following best describes the term v?

Answer 8

the ratio of displacement of an object to time required for the displacement

Question 9

The displacement of a particle is shown in the plot. At which of the following points is the instantaneous velocity 0 m / s?

Answer 9

t2, t4, and t6

Question 10

Look at the plot that shows the position of a falling ball as a function of time, t. Which of the following statements about this event is not correct?

Answer 10

The greatest magnitude of the average velocity occurs from t1 to t2.

Question 11

Given a graph of position versus time, which of the following is correct regarding the relationships between the secant line, the tangent line, and the values for average and instantaneous velocity?

Answer 11

When the change in time approaches 0, the slope of the secant line is equal to the slope of the tangent line, which is equal to the magnitude of the average velocity, which is equal to the magnitude of the instantaneous velocity.