Motion in a Straight Line Flashcards

Comprehensive vocabulary flashcards covering the basic and advanced concepts of kinematics including speed, velocity, acceleration, and graphical analysis.

Speed

The rate at which distance is covered with respect to time; it is a scalar quantity.

S.I. Unit of Speed

$m/s$ (Metre per second).

Dimension of Speed

$[M^0 L^1 T^{-1}]$.

Conversion formula for km/hr to m/s

Multiply the value by $\frac{5}{18}$.

Uniform Speed

A condition where a particle covers equal distances in equal intervals of time.

Non-uniform Speed (Variable Speed)

A condition where a particle covers unequal distances in equal intervals of time.

Average Speed

The ratio of total distance travelled to total time taken, expressed as $V_{avg} = \frac{\text{Total distance travelled}}{\text{Total time taken}} = \frac{\triangle s}{\triangle t}$.

Instantaneous Speed

The speed of a particle at a particular instant of time, defined as $V_{inst} = \text{lim}_{\triangle t \rightarrow 0} \frac{\triangle s}{\triangle t} = \frac{ds}{dt}$.

Average Speed (First half with $v_1$, next half with $v_2$)

$V_{avg} = \frac{2v_1 v_2}{v_1 + v_2}$.

Arithmetic Mean of Individual Speeds

Occurs when a particle travels with speeds $v_1$, $v_2$, etc., during equal time intervals ($t_1 = t_2 = ... = t$), calculated as $V_{avg} = \frac{v_1 + v_2 + ... + v_n}{n}$.

Velocity

The rate of change of position with time; it is a vector quantity.

Uniform Velocity (Constant Velocity)

Velocity that remains the same in both magnitude and direction, occurring only when moving in a straight line without reversing.

Average Velocity

The ratio of net displacement to time taken, expressed as $V_{avg} = \frac{\text{Net Displacement}}{\text{Time taken}} = \frac{\triangle \textbf{r}}{\triangle t}$.

Instantaneous Velocity

The velocity of a particle at a particular instant of time, defined as $V_{inst} = \frac{d\textbf{r}}{dt}$, always tangential to the path followed.

Uniform Circular Motion (Speed and Velocity relationship)

A case where speed remains constant but velocity changes at every instant due to change in direction.

Acceleration

The rate of change of velocity; it is a vector quantity with direction same as the change in velocity.

S.I. Unit of Acceleration

$m/s^2$ (Metre per second square).

Dimension of Acceleration

$[M^0 L^1 T^{-2}]$.

Uniform Acceleration

Motion where the magnitude and direction of acceleration remain constant.

Average Acceleration

The ratio of total change in velocity to the total time taken: $a_{avg} = \frac{\triangle v}{\triangle t} = \frac{v_2 - v_1}{t_2 - t_1}$.

Instantaneous Acceleration

The first derivative of velocity with respect to time ($a = \frac{dv}{dt}$) or the second derivative of the position vector ($a = \frac{d^2 \textbf{r}}{dt^2}$).

Slope of Position-time (x-t) Graph

Represents the instantaneous velocity ($\text{tan}(\theta) = \frac{dx}{dt} = \text{velocity}$).

Slope of Velocity-time (v-t) Graph

Represents the acceleration ($\text{tan}(\theta) = \frac{\triangle v}{\triangle t} = \text{acceleration}$).

Area under v-t Graph

Gives displacement if signs are considered (Area 1 - Area 2) and distance if magnitudes are summed (Area 1 + Area 2).

Area under a-t Graph

Represents the change in velocity ($v_f - v_i$).

Velocity as a function of time (Kinematical Equation)

$v = u + at$.

Displacement as a function of time (Kinematical Equation)

$s = ut + \frac{1}{2}at^2$.

Velocity as a function of displacement (Kinematical Equation)

$v^2 - u^2 = 2as$.

Displacement in the $n^{th}$ second of motion

$S_n = u + \frac{a}{2}(2n - 1)$.

Maximum Velocity for combined Acceleration and Retardation

When a car starts from rest, accelerates at $\alpha$, retards at $\beta$, and comes to rest in total time $T$, the maximum velocity is V_{max} = \frac{\text{\alpha\beta}}{(\text{\alpha + \beta})} T.