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Q3.28(f):
The velocity-time graph of a particle in one-dimensional motion is shown in Fig. 3.29 : Which of the following formulae are correct for describing the motion of the particle over the time-interval $t_1$ to $t_2$: (f) $x(t_2) - x(t_1)$ = area under the $v-t$ curve bounded by the $t$-axis and the dotted line shown.
The velocity-time graph of a particle in one-dimensional motion is shown in Fig. 3.29 : Which of the following formulae are correct for describing the motion of the particle over the time-interval $t_1$ to $t_2$: (f) $x(t_2) - x(t_1)$ = area under the $v-t$ curve bounded by the $t$-axis and the dotted line shown.
Solution :
Step 1: Theoretical Foundation of Velocity-Time Graphs
The motion of a particle in one dimension is defined by the instantaneous velocity $v(t)$, which is the derivative of position $x(t)$ with respect to time $t$: $v(t) = \frac{dx}{dt}$ [Per the definition of instantaneous velocity].
Step 2: Deriving the Integral Form
To determine the change in position (displacement) over a time interval $[t_1, t_2]$, we rearrange the differential equation: $dx = v(t) dt$.
Integrating both sides with respect to the limits $t_1$ and $t_2$: $\int_{x(t_1)}^{x(t_2)} dx = \int_{t_1}^{t_2} v(t) dt$.
This yields: $x(t_2) - x(t_1) = \int_{t_1}^{t_2} v(t) dt$. [By the Fundamental Theorem of Calculus].
Step 3: Geometric Interpretation
In a Cartesian coordinate system where the ordinate is $v$ and the abscissa is $t$, the definite integral $\int_{t_1}^{t_2} v(t) dt$ represents the net signed area enclosed between the curve $v(t)$, the horizontal axis ($t$-axis), and the vertical lines $t = t_1$ and $t = t_2$.
Visual Representation of Displacement
Step 4: Analytical Validation
The expression $x(t_2) - x(t_1)$ represents the displacement of the particle between $t_1$ and $t_2$. Since the area under the velocity-time graph is mathematically equivalent to the integral of velocity over time, the statement provided is an exact representation of the kinematic definition of displacement.
| Physical Quantity | Mathematical Representation |
|---|---|
| Displacement | $\Delta x = x(t_2) - x(t_1)$ |
| Area under $v-t$ curve | $\int_{t_1}^{t_2} v(t) dt$ |
Final Solution: The formula $x(t_2) - x(t_1) = \int_{t_1}^{t_2} v(t) dt$ is correct, as the displacement of a particle is identically defined by the area bounded by the velocity-time graph, the time axis, and the time interval boundaries.
More Questions from Class 11 Physics Motion in a Straight Line Additional Exercises
- Q3.23: A three-wheeler starts from rest, accelerates uniformly with $1 m s^{-2}$ on a straight road for $10 s$, and then moves with uniform velocity. Plot the distance covered by the vehicle during the nth second ($n$ = 1,2,3….) versus $n$. What do you expect this plot to be during accelerated motion : a straight line or a parabola ?
- Q3.24: A boy standing on a stationary lift (open from above) throws a ball upwards with the maximum initial speed he can, equal to $49 m s^{-1}$. How much time does the ball take to return to his hands? If the lift starts moving up with a uniform speed of $5 m s^{-1}$ and the boy again throws the ball up with the maximum speed he can, how long does the ball take to return to his hands ?
- Q3.25(a): On a long horizontally moving belt (Fig. 3.26), a child runs to and fro with a speed $9 km h^{-1}$ (with respect to the belt) between his father and mother located $50 m$ apart on the moving belt. The belt moves with a speed of $4 km h^{-1}$. For an observer on a stationary platform outside, what is the (a) speed of the child running in the direction of motion of the belt ?.
- Q3.25(b): On a long horizontally moving belt (Fig. 3.26), a child runs to and fro with a speed $9 km h^{-1}$ (with respect to the belt) between his father and mother located $50 m$ apart on the moving belt. The belt moves with a speed of $4 km h^{-1}$. For an observer on a stationary platform outside, what is the (b) speed of the child running opposite to the direction of motion of the belt ?
- Q3.25(c): On a long horizontally moving belt (Fig. 3.26), a child runs to and fro with a speed $9 km h^{-1}$ (with respect to the belt) between his father and mother located $50 m$ apart on the moving belt. The belt moves with a speed of $4 km h^{-1}$. For an observer on a stationary platform outside, what is the (c) time taken by the child in (a) speed of the child running in the direction of motion of the belt ? and (b) speed of the child running opposite to the direction of motion of the belt ? Which of the answers alter if motion is viewed by one of the parents ?
- Q3.26: Two stones are thrown up simultaneously from the edge of a cliff $200 m$ high with initial speeds of $15 m s^{-1}$ and $30 m s^{-1}$. Verify that the graph shown in Fig. 3.27 correctly represents the time variation of the relative position of the second stone with respect to the first. Neglect air resistance and assume that the stones do not rebound after hitting the ground. Take $g = 10 m s^{-2}$. Give the equations for the linear and curved parts of the plot.
- Q3.27(a): The speed-time graph of a particle moving along a fixed direction is shown in Fig. 3.28. Obtain the distance traversed by the particle between (a) $t = 0 s$ to $10 s$.
- Q3.27(b): The speed-time graph of a particle moving along a fixed direction is shown in Fig. 3.28. Obtain the distance traversed by the particle between (b) $t = 2 s$ to $6 s$.
- Q3.27(c): The speed-time graph of a particle moving along a fixed direction is shown in Fig. 3.28. What is the average speed of the particle over the interval in (a) $t = 0 s$ to $10 s$?
- Q3.27(d): The speed-time graph of a particle moving along a fixed direction is shown in Fig. 3.28. What is the average speed of the particle over the interval in (b) $t = 2 s$ to $6 s$?
- Q3.28(a): The velocity-time graph of a particle in one-dimensional motion is shown in Fig. 3.29 : Which of the following formulae are correct for describing the motion of the particle over the time-interval $t_1$ to $t_2$: (a) $x(t_2) = x(t_1) + v(t_1) (t_2 - t_1) +(\frac{1}{2}) a (t_2 - t_1)^2$
- Q3.28(b): The velocity-time graph of a particle in one-dimensional motion is shown in Fig. 3.29 : Which of the following formulae are correct for describing the motion of the particle over the time-interval $t_1$ to $t_2$: (b) $v(t_2) = v(t_1) + a (t_2 - t_1)$
- Q3.28(c): The velocity-time graph of a particle in one-dimensional motion is shown in Fig. 3.29 : Which of the following formulae are correct for describing the motion of the particle over the time-interval $t_1$ to $t_2$: (c) $v_{average} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}$
- Q3.28(d): The velocity-time graph of a particle in one-dimensional motion is shown in Fig. 3.29 : Which of the following formulae are correct for describing the motion of the particle over the time-interval $t_1$ to $t_2$: (d) $a_{average} = \frac{v(t_2) - v(t_1)}{t_2 - t_1}$
- Q3.28(e): The velocity-time graph of a particle in one-dimensional motion is shown in Fig. 3.29 : Which of the following formulae are correct for describing the motion of the particle over the time-interval $t_1$ to $t_2$: (e) $x(t_2) = x(t_1) + v_{average} (t_2 - t_1) + (\frac{1}{2}) a_{average} (t_2 - t_1)^2$
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