Question

1. for each of the position vs time graphs shown below, draw the corresponding v vs t, a vs t, and motion map.

Explanation:

Step1: Recall derivative relationships

Velocity $v$ is the derivative of position $x$ with respect to time $t$ ($v=\frac{dx}{dt}$), and acceleration $a$ is the derivative of velocity $v$ with respect to time $t$ ($a = \frac{dv}{dt}=\frac{d^{2}x}{dt^{2}}$).

Step2: Analyze first position - time graph

The position - time graph is a decreasing linear part followed by a flat part. For the decreasing linear part, the slope is negative and constant, so the velocity is negative and constant. For the flat part, the slope is 0, so the velocity is 0. Since the velocity is constant in the first part and 0 in the second part, the acceleration is 0 throughout.

Step3: Analyze second position - time graph

The position - time graph is a parabola opening downwards. The slope of the position - time graph (velocity) is initially positive, decreases to 0 at the maximum of the parabola, and then becomes negative. The acceleration is negative and constant because the rate of change of velocity is constant.

Step4: Analyze third position - time graph

The position - time graph is an increasing non - linear curve. The slope of the position - time graph (velocity) is positive and increasing. So the acceleration is positive and non - zero.

Answer:

For the first set of graphs:

• Velocity - time graph: Negative constant value for the first part of time, 0 for the second part.
• Acceleration - time graph: 0 throughout.
• Motion map: Arrows for velocity are of constant length and direction (left - pointing) for the first part and 0 for the second part; no acceleration arrows (since $a = 0$).

For the second set of graphs:

• Velocity - time graph: Positive and decreasing to 0 and then negative.
• Acceleration - time graph: Negative and constant.
• Motion map: Velocity arrows are initially long and right - pointing, get shorter until 0 at the peak of the motion, and then become long and left - pointing; acceleration arrows are of constant length and left - pointing.

For the third set of graphs:

• Velocity - time graph: Positive and increasing.
• Acceleration - time graph: Positive and non - zero.
• Motion map: Velocity arrows are getting longer and right - pointing; acceleration arrows are of constant length and right - pointing.