Analyzing Motion Using Graphs


[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:


b. When does it move to the left (down) or to the right (up)?


s = 200t - 16t², 0 ≤ t ≤ 12.5 (a heavy object fired straight up from Earth’s surface at 200 ft/sec)

Verified step by step guidance

1. First, identify the position function given: s(t) = 200t - 16t². This function describes the position of the object over time.
2. To find the velocity function v(t), take the first derivative of the position function with respect to time t. This gives v(t) = ds/dt = f'(t).
3. Calculate the first derivative: v(t) = d/dt [200t - 16t²] = 200 - 32t. This represents the velocity of the object as a function of time.
4. To find the acceleration function a(t), take the derivative of the velocity function with respect to time t. This gives a(t) = dv/dt = f''(t).
5. Calculate the second derivative: a(t) = d/dt [200 - 32t] = -32. This constant acceleration indicates the object is subject to a constant deceleration due to gravity.

Key Concepts

Position, Velocity, and Acceleration

In calculus, the position function s(t) describes the location of an object over time. The velocity function v(t) is the first derivative of the position function, representing the rate of change of position, or speed and direction of motion. Acceleration a(t) is the second derivative of the position function, indicating the rate of change of velocity. Understanding these relationships is crucial for analyzing motion.

Derivatives and Their Physical Interpretation

The derivative of a function at a point gives the slope of the tangent line to the function at that point, representing the rate of change. In the context of motion, the first derivative (velocity) indicates how fast and in what direction the position changes, while the second derivative (acceleration) shows how the velocity changes over time. These interpretations help in understanding the object's movement.

Sign of Velocity and Acceleration

The sign of the velocity function v(t) indicates the direction of motion: positive for right/upward and negative for left/downward. The sign of the acceleration function a(t) shows whether the object is speeding up or slowing down. Positive acceleration means the velocity is increasing, while negative acceleration means it is decreasing. Analyzing these signs helps determine the object's behavior over time.