Question

Suppose the motion of the particle is described by the equation: x=a+bt2, where a=20cm and b=4cm.s-2. a) Find the displacement of the particle in the time interval between t1 =2s and t2 =5s. b) Find the average velocity in this time interval. C) Find the instantaneous velocity at time t1 =2s.

Answer 1

Suppose the motion of the particle is described by the equation: $x = a + bt2$ , where $a = 20\mathrm{cm}$ and $b = 4\mathrm{cm}$ .s-2. a) Find the displacement of the particle in the time interval between t1 =2s and t2 =5s. b) Find the average velocity in this time interval. C) Find the instantaneous velocity at time t1 =2s.

Solution: the equation of motion for a particle and the equation of motion in the general form:

$x = a + bt^{2},\qquad \text{general:} x = x_{0} + V_{0}t + \frac{At^{2}}{2},\text{where} A - \text{acceleration},V_{0} - \text{initial velocity}$

Comparing the two equations we can find the initial speed, position and the acceleration:

$x_{0} = a, V_{0} = 0, A = 2b$

A: Because the coordinate of the particle is increasing, we will find the position of a particle at point 1 and point 2 and subtract the values of the coordinates. The difference will be equal to the displacement of the particle.

$$x _ {t 1} = a + b t _ {1} ^ {2}, \quad x _ {t 2} = a + b t _ {2} ^ {2}$$ $$\Delta S = x _ {t 2} - x _ {t 1} = a + b t _ {2} ^ {2} - a - b t _ {1} ^ {2} = b \left(t _ {2} ^ {2} - t _ {1} ^ {2}\right) = 4 \frac {c m}{s ^ {2}} * (5 s * 5 s - 2 s * 2 s) = 8 4 c m$$

B: Average speed is equal to the length of way divided by the time in which the particle passes this way:

$$V _ {a} = \frac {\Delta S}{\Delta t}$$ $$\Delta t = t _ {2} - t _ {1}$$ $$V _ {a} = \frac {\Delta S}{\Delta t} = \frac {8 4 c m}{5 s - 2 s} = 2 8 \frac {c m}{s}$$

C: The equation of motion for a particle:

$$V = V _ {0} + A t, \qquad V _ {0} = 0, \qquad A = 2 b = 8 \frac {c m}{s ^ {2}}$$

Instantaneous velocity at time $t = 2s$ :

$$V _ {i} = A t = 8 \frac {c m}{s ^ {2}} * 2 s = 1 6 \frac {c m}{s}$$

Answer: a) $84 \, \text{cm}$ b) $28 \frac{\text{cm}}{\text{s}}$ , c) $16 \frac{\text{cm}}{\text{s}}$ .