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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 ?.
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 ?.
Solution :
Step 1: Establishing Coordinate Systems and Variables
To determine the velocity of the child with respect to a stationary observer, we apply the principles of relative velocity in one dimension [Per the Galilean Transformation of velocities].
- Velocity of the belt relative to the stationary ground ($v_{b}$): $+4 \text{ km h}^{-1}$
- Velocity of the child relative to the belt ($v_{c/b}$): $+9 \text{ km h}^{-1}$ (running in the direction of the belt's motion)
- Velocity of the child relative to the stationary ground ($v_{c}$): To be determined.
Step 2: Vector Addition of Relative Velocities
The velocity of an object $c$ with respect to a stationary frame $g$ is the vector sum of its velocity with respect to a moving frame $b$ and the velocity of that moving frame with respect to the stationary frame $g$. [By the Law of Addition of Velocities]:
$v_{c} = v_{c/b} + v_{b}$
Step 3: Algebraic Calculation
Substitute the given values into the equation:
$v_{c} = 9 \text{ km h}^{-1} + 4 \text{ km h}^{-1}$
$v_{c} = 13 \text{ km h}^{-1}$
Visual Representation of Vector Addition
Step 4: Verification and Unit Consistency
The calculation assumes the motion is strictly linear and collinear. Since both the belt and the child move in the positive direction relative to the ground, the magnitudes are added arithmetically.
| Component | Value |
|---|---|
| $v_{c/b}$ | $9 \text{ km h}^{-1}$ |
| $v_{b}$ | $4 \text{ km h}^{-1}$ |
| $v_{c}$ (Resultant) | $13 \text{ km h}^{-1}$ |
Final Solution: The speed of the child running in the direction of motion of the belt, as observed from the stationary platform, is $13 \text{ km h}^{-1}$.
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(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) $x(t_2) = x(t_1) + v_{average} (t_2 - t_1) + (\frac{1}{2}) a_{average} (t_2 - t_1)^2$
- 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.
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