Particle Position Calculation
The position of the particle is given by the function $s(t) = \alpha t^2 - \beta t + \gamma$. The provided constants are $\alpha = 1\text{ ms}^{-2}$, $\beta = 6\text{ ms}^{-1}$, and $\gamma = 5\text{ m}$.
First, find the position at the start time, $t = 0\text{ s}$:
$s(0) = (1)(0)^2 - (6)(0) + 5 = 5\text{ m}$
Next, find the position at the end time, $t = 6\text{ s}$:
$s(6) = (1)(6)^2 - (6)(6) + 5 = 36 - 36 + 5 = 5\text{ m}$
Velocity and Direction Change Check
To determine the total distance traveled, we need to check if the particle changed direction. This occurs when the velocity, $v(t)$, is zero.
The velocity function is the derivative of the position function:
$v(t) = s'(t) = \frac{d}{dt}(\alpha t^2 - \beta t + \gamma) = 2\alpha t - \beta$
Substitute the values for $\alpha$ and $\beta$:
$v(t) = 2(1)t - 6 = 2t - 6\text{ ms}^{-1}$
Set the velocity to zero to find the time(s) of direction change:
$2t - 6 = 0 \implies 2t = 6 \implies t = 3\text{ s}$
Since $t = 3\text{ s}$ falls within the interval $[0, 6\text{ s}]$, the particle does change direction.
Distance Traveled Calculation
Calculate the position at the time of direction change, $t = 3\text{ s}$:
$s(3) = (1)(3)^2 - (6)(3) + 5 = 9 - 18 + 5 = -4\text{ m}$
Now, calculate the distance traveled during each part of the motion:
- Distance from $t=0$ to $t=3\text{ s}$: The particle moves from $s(0)=5\text{ m}$ to $s(3)=-4\text{ m}$. The distance is $|s(3) - s(0)| = |-4\text{ m} - 5\text{ m}| = |-9\text{ m}| = 9\text{ m}$.
- Distance from $t=3$ to $t=6\text{ s}$: The particle moves from $s(3)=-4\text{ m}$ to $s(6)=5\text{ m}$. The distance is $|s(6) - s(3)| = |5\text{ m} - (-4\text{ m})| = |9\text{ m}| = 9\text{ m}$.
The total distance traveled is the sum of these distances:
$ \text{Total Distance} = 9\text{ m} + 9\text{ m} = 18\text{ m} $
Average Speed Calculation
The total time elapsed is $T = 6\text{ s} - 0\text{ s} = 6\text{ s}$.
Average speed is defined as the total distance traveled divided by the total time taken:
$ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{18\text{ m}}{6\text{ s}} = 3\text{ ms}^{-1} $
Therefore, the average speed of the particle is $3\text{ ms}^{-1}$.
Similar Questions
- The magnitude and direction of the acceleration produced in a body of mass $5 \text{ kg}$ when two mutually perpendicular forces $8 \text{ N}$ and $6 \text{ N}$ act on it, are respectively :
- A box of mass $15 \text{ kg}$ is kept on the floor of a stationary trolley. The coefficient of static friction between the box and the trolley is $0.12$. Keeping the box in stationary state over the trolley, the maximum acceleration with which the trolley can be moved horizontally is : ($g = 10 \text{ m/s}^2$)
- The amount of work done to raise a mass 'm' from the surface of the Earth to a height equal to the radius of the Earth 'R', will be :
A thin wire of length 'L' and linear mass density 'm' is bent into a circular ring (in x-y plane) with centre 'C' as shown in figure. The moment of inertia of the ring about an axis yy' (tangent in the plane) will be :
- When a ruler falls vertically, 5 different persons catch it with different reaction times. ($g = 9.8 \text{ m s}^{-2}$)
A. Person A has reaction time of $0.20 \text{ s}$.
B. Person B has reaction time of $0.22 \text{ s}$.
C. Person C has reaction time of $0.18 \text{ s}$.
D. Person D has reaction time of $0.19 \text{ s}$.
E. Person E has reaction time of $0.21 \text{ s}$.
What is the correct order of the distance travelled by the ruler for each person ? - The angular speed of a flywheel is increased from $600 \text{ rpm}$ to $1200 \text{ rpm}$ in $10 \text{ s}$. The number of revolutions completed by the flywheel during this time is :
- A particle of mass M moves along a horizontal x axis from $x=0$ to $x=L$. The coefficient of kinetic friction varies as a function of $x$ as $\mu_k(x) = \mu_0 - \alpha x$, where $\mu_0$, $\alpha$ are constants of appropriate dimensions, so that $\mu_k(L) = 0$. The total work done by the frictional force during the motion is $n \mu_0 M g L$, where g is the acceleration due to gravity. The value of $n$ is :
- In a solar system, the time-period of revolution of a planet tracing a circular orbit of radius $R$ is proportional to :
Bob B of mass $m$ at rest is hanging vertically from the ceiling via a massless string of length $10\text{ m}$, as shown in the figure. Point mass A of mass $m$ travelling horizontally with speed $10\text{ ms}^{-1}$ hits bob B elastically. The bob B rises $h$ meter after the collision. Taking the acceleration due to gravity $g = 10\text{ ms}^{-2}$ and neglecting the size of the bob, the value of $h$ is :
A thin horizontal disc is rotating about a vertical axis passing through its fixed centre O. Its angular momentum is $L_A$ and $L_B$ computed about points A and B, respectively, with $OB = 2 \times OA$. The value of $\frac{L_A}{L_B}$ is :
Important Questions from Mechanics
- When both jaws of vernier callipers touch each other, zero mark of the vernier scale is right to zero mark of main scale, $4^{\text{th}}$ mark on vernier scale coincides with certain mark on the main scale. While measuring the length of a cylinder, observer observes $15$ divisions on main scale and $5^{\text{th}}$ division of vernier scale coincides with a main scale division. Measured length of cylinder is ________mm.
(Least count of Vernier calliper = $0.1 \text{ mm}$) - Water drops fall from a tap on the floor, $5 \text{ m}$ below, at regular intervals of time, the first drop strikes the floor when the sixth drop begins to fall. The height at which the fourth drop will be from ground, at the instant when the first drop strikes the ground is ________m.
($g = 10 \text{ m/s}^2$) - Two wires $A$ and $B$ made of different materials of lengths $6.0 \text{ cm}$ and $5.4 \text{ cm}$, respectively and area of cross sections $3.0 \times 10^{-5} \text{ m}^2$ and $4.5 \times 10^{-5} \text{ m}^2$, respectively are stretched by the same magnitude under a given load. The ratio of the Young's modulus of $A$ to that of $B$ is $x : 3$. The value of $x$ is________.
Two circular discs of radius each $10 \text{ cm}$ are joined at their centres by a rod of length $30 \text{ cm}$ and mass $600 \text{ gm}$ as shown in figure.
If the mass of each disc is $600 \text{ gm}$ and applied torque between two discs is $43 \times 10^5 \text{ dyne.cm}$, the angular acceleration of the discs about the given axis $AB$ is________$\text{rad/s}^2$.
- A particle of mass $m$ falls from rest through a resistive medium having resistive force, $F = -kv$, where $v$ is the velocity of the particle and $k$ is a constant. Which of the following graphs represents velocity ($v$) versus time ($t$)?
