**Goal:** Relate acceleration to slope of velocity graph

**Source:** UMPERG

A toy rocket blasts off with an acceleration of 9.8 m/s^{2}

upward. After 5 seconds the rocket releases its “payload” but continues

to accelerate upward at the same rate. In the following graph the solid

line represents the velocity of the rocket as a function of time and the

dashed line represents the velocity of the payload.

Which of the following statements regarding this situation are correct?

A. The payload hits the ground at t = 10s.

B. The slope of the rocket’s velocity vs. time graph is 9.8 m/s

^{2}.C. The velocities of the rocket and the payload point in opposite

directions after the payload is released.D. The area of the shaded region can be determined from the given

information.E. The payload spends half the time in the air as the rocket.

F. The payload is released at 1/2 the maximum height of the rocket.

- Only (A)
- Only (B)
- Only (C)
- Only (D)
- Only (E)
- Only (F)
- Two of the statements are true
- Three of the statements are true
- None is true

## Commentary:

## Answer

Answer (7) is the best choice. The only statements that are true are

(B) and (D). The height of the rocket in relation to the height of the

payload can be determined by the ratio of the areas under their

respective velocity vs. time graphs. The payload reaches its maximum

height at t=10s (i.e., when its velocity is equal to zero). At t=10s the

area under the rocket’s v vs. t graph is twice the area under payload’s

v vs. t graph. This can be determined without knowing any of the

information contained in (A) through (F).

## Background

Issues to consider: (1) Can students identify and evaluate information

needed to judge the correctness of a statement? (2) Can students

interpret a graphical representation of information? (3) Can students

determine the point of maximum displacement from a v vs. t graph? (4)

Can students interpret the significance of slope and area for a v vs. t

graph. (5) Do students confuse v vs. t graphs with x vs. t graphs.

## Questions to Reveal Student Thinking

When does the payload start to fall back toward the earth? When does

the payload hit the earth? What is the acceleration of the payload

after it is released?

Which of the statements are true? Explain. Which of the statements are

false? Explain.

Could the true statements be used to determine the heights of the rocket

and payload? Explain.

## Suggestions

As a class draw a rough strobe diagram. Relate times in the strobe

diagram to times on the graph.

Discuss with students how they would approach the problem algebraically.