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Are my graphs and equations right? I just began learning about these..

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    $\begingroup$ You need constants of integration: $\int 2\,dt = 2t + c$ and so on. $\endgroup$
    – user107153
    Jan 28, 2019 at 14:12
  • $\begingroup$ Or you need the initial and final value of $t$. $\int_{t_i}^{t_f} 2 = 2 (t_f-t_i)$ $\endgroup$ Jan 28, 2019 at 14:14

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Not in general. You're forgetting all of the constants of integration.

Suppose you start with a constant jerk $j$. Then the acceleration $a$ as a function of time is

$$a(t)=\int j\;dt=jt+a_0$$

where $a_0$ is the initial acceleration. This means that the velocity $v$ as a function of time is

$$v(t)=\int a(t)\;dt=\int (jt+a_0)\;dt=\frac{1}{2}jt^2+a_0t+v_0$$

where $v_0$ is the initial velocity. Likewise, the position $x$ as a function of time will be

$$x(t)=\int v(t)\;dt=\int\left(\frac{1}{2}jt^2+a_0t+v_0\right)\;dt=\frac{1}{6}jt^3+\frac{1}{2}a_0t^2+v_0t+x_0$$

where $x_0$ is the initial position. So, for motion with constant jerk to be well-defined, you need to specify an initial position, velocity, and acceleration.

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