Speed-Time Graphs

Problem 1. The speed-time graph of a particle moving in a straight line is shown below.

On separate diagrams, plot the corresponding distance-time graph and acceleration-time graph of the particle.

(Click for Solution)

Solution. Let $s(t)$ denote the total distance travelled by the particle at time $t$ . Using the area under the graph,

$s(0) = 0,\quad s(6) = 18,\quad s(10) = 42,\quad s(14) = 54.$

Therefore, we draw the distance-time graph as follows.

Let $a(t)$ denote the acceleration of the particle at time $t$ . Using the gradient of the graph at various sample points,

$a(2) = 1,\quad a(8) = 0,\quad a(12) = -1.5.$

Therefore, we draw the acceleration-time graph as follows.

Remark 1. Using calculus, the shape of $s(t)$ for $0 \leq t \leq 6$ is given by

$\displaystyle \int_0^t s(x)\, \mathrm dx = \int_0^t x\, \mathrm dx = \frac{t^2}{2}.$

Problem 2. Newton’s second law states that the (net) force acting on an object is defined by the rate of change of its momentum.

The force-time graph for an object (initially at rest) is shown below.

Plot the momentum-time graph of the object.

(Click for Solution)

Solution. Let $p(t)$ denote the total momentum of the object at time $t$ . Using the area between the graph and the Time axis,

$p(0) = 0,\quad p(3) = 6,\quad p(5) = 6,\quad p(8) = -3.$

Therefore, we draw the momentum-time graph as follows.

—Joel Kindiak, 9 Apr 26, 0017H

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