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 power of a crane, which lifts a mass of $1000 \text{ kg}$ to a height of $20 \text{ m}$ in $10 \text{ s}$ is :
($g = 9.8 \text{ m/s}^2$)The following plots show variation of velocity ($v$) with time ($t$) of a ball thrown vertically upward, and falling back. Which of the following plots is/are correct?
- 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 frictionless circular wire of unit radius is fixed on the horizontal plane. Two point particles of unit mass start moving simultaneously from point $A \left( \theta = \frac{\pi}{2} \right)$ with identical uniform angular speeds in opposite directions, and meet again at point $B \left( \theta = -\frac{\pi}{2} \right)$. During this time, which of the following figures schematically represent the magnitude of the total linear momentum $\vec{P}$ of the system, as a function of $\theta$?
- Two planets $P_1$ and $P_2$ with equal mass have radii $R_1$ and $R_2$, respectively, where $R_2 = \frac{R_1}{2}$. The escape speeds of $P_1$ and $P_2$ are $v_1$ and $v_2$, respectively. Then $\frac{v_2}{v_1}$ is :
Important Questions from Mechanics
- A soap bubble of surface tension $0.04 \text{ N/m}$ is blown to a diameter of $7 \text{ cm}$. If $(15000 - x) \text{ }\mu\text{J}$ of work is done in blowing it further to make its diameter $14 \text{ cm}$, then the value of $x$ is _______.
($\pi = 22/7$) - A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L/2$ and radius $R/3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1/I_2$ is _______.
- A fly wheel having mass $3\text{ kg}$ and radius $5\text{ m}$ is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to $3\text{ kg}$ mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by $3\text{ m}$ is _________ $\text{J}$. ($g = 10\text{ m/s}^{2}$)
Suppose there is a uniform circular disc of mass $M \text{ kg}$ and radius $r \text{ m}$ shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis $A$ of the disc is given by $\frac{x}{256} M r^2$. The value of $x$ is ________.
- A ball of radius $r$ and density $\rho$ dropped through a viscous liquid of density $\sigma$ and viscosity $\eta$ attains its terminal velocity at time $t$, given by $t = A \rho^a r^b \eta^c \sigma^d$, where $A$ is a constant and $a$, $b$, $c$ and $d$ are integers. The value of $\frac{b+c}{a+d}$ is ________.
