Question

sph3u - unit 1 kinematics

1. convert the following position - time graph into a velocity time graph, and an acceleration time graph.

4 marks - c
kinematic equations:
$vec{delta d}=\frac{vec{v_i}+vec{v_f}}{2}cdotdelta t$
$vec{v_f}=vec{v_i}+vec{a}delta t$
$vec{delta d}=vec{v_i}delta t+\frac{1}{2}vec{a}delta t^{2}$
$vec{delta d}=vec{v_f}delta t-\frac{1}{2}vec{a}delta t^{2}$
$vec{v_f}^{2}=vec{v_i}^{2}+2vec{a}delta d$
other useful equations:
$d = vdelta t$
$vec{v_{avg}}=\frac{vec{delta d}}{delta t}$
$vec{a}=\frac{vec{delta v}}{delta t}$
$m=\frac{rise}{run}=\frac{y_2 - y_1}{x_2 - x_1}$

Explanation:

Step1: Recall velocity - position relationship

Velocity $v$ is the slope of the position - time graph $d(t)$. Mathematically, $v=\frac{\Delta d}{\Delta t}$. When the position - time graph is increasing, the slope (velocity) is positive. When it is decreasing, the slope (velocity) is negative. And when the graph is flat (horizontal), the slope (velocity) is zero.

Step2: Analyze the given position - time graph

1. At the start, the position - time graph is increasing, so the velocity is positive.
2. As the graph reaches its maximum point, the slope is zero, so the velocity is zero at that instant.
3. After the maximum point, the graph starts to decrease, so the velocity is negative.
4. As the graph becomes horizontal at some points during its decrease, the velocity is zero at those points.

Step3: Recall acceleration - velocity relationship

Acceleration $a$ is the slope of the velocity - time graph $v(t)$. Mathematically, $a = \frac{\Delta v}{\Delta t}$. When the velocity - time graph is increasing, the slope (acceleration) is positive. When it is decreasing, the slope (acceleration) is negative. And when the graph is flat (horizontal), the slope (acceleration) is zero.

Step4: Analyze the velocity - time graph to get acceleration - time graph

1. When the velocity is increasing (going from zero to positive or from a negative value to a less - negative or positive value), the acceleration is positive.
2. When the velocity is decreasing (going from positive to zero or from a positive value to a more negative value), the acceleration is negative.
3. When the velocity is constant (horizontal part of the velocity - time graph), the acceleration is zero.

To actually draw the graphs:

1. For the velocity - time graph:

• Mark the time intervals where the position - time graph is increasing as positive - velocity regions on the velocity - time graph.
• Mark the time intervals where the position - time graph is decreasing as negative - velocity regions on the velocity - time graph.
• Mark the time instants where the position - time graph has a horizontal tangent as zero - velocity points on the velocity - time graph.

1. For the acceleration - time graph:

• Mark the time intervals where the velocity - time graph is increasing as positive - acceleration regions on the acceleration - time graph.
• Mark the time intervals where the velocity - time graph is decreasing as negative - acceleration regions on the acceleration - time graph.
• Mark the time intervals where the velocity - time graph is horizontal as zero - acceleration regions on the acceleration - time graph.

Since we can't draw the actual graphs here in a visual sense, the above steps explain the process of converting the position - time graph to a velocity - time graph and then to an acceleration - time graph.

Answer:

The process to convert the position - time graph to a velocity - time graph and then to an acceleration - time graph is as described in the steps above. To obtain the actual graphs, follow the rules of slope - calculation for velocity (slope of position - time graph) and acceleration (slope of velocity - time graph) at different time intervals of the given position - time graph.