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QA1(a):
The following is the distance-time table of an object in motion:
(a) What conclusion can you draw about the acceleration? Is it constant, increasing, decreasing, or zero?
The following is the distance-time table of an object in motion:
(a) What conclusion can you draw about the acceleration? Is it constant, increasing, decreasing, or zero?
Solution :
Step 1: Analyzing the Provided Data
Given the distance ($s$) and time ($t$) values provided in the table:
| Time ($s$) | Distance ($m$) |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
Step 2: Mathematical Evaluation of the Motion
To determine the nature of the acceleration, we evaluate the relationship between distance ($s$) and time ($t$). Observing the numerical pattern:
- At $t=1, s=1^3=1$
- At $t=2, s=2^3=8$
- At $t=3, s=3^3=27$
- At $t=4, s=4^3=64$
The motion follows the cubic function: $s(t) = kt^3$, where $k=1 \text{ m/s}^3$. [Based on polynomial regression analysis of the data points].
Step 3: Deriving Velocity and Acceleration
To find velocity ($v$), we calculate the first derivative of distance with respect to time [Per the definition of instantaneous velocity, $v = \frac{ds}{dt}$]:
$v = \frac{d}{dt}(t^3) = 3t^2$
To find acceleration ($a$), we calculate the derivative of velocity with respect to time [Per the definition of instantaneous acceleration, $a = \frac{dv}{dt}$]:
$a = \frac{d}{dt}(3t^2) = 6t$
Step 4: Visualizing the Motion Profile
Step 5: Conclusion on Acceleration
Since the derived expression for acceleration is $a = 6t$, the value of acceleration is directly proportional to time ($t$). As $t$ increases, $a$ increases linearly. Therefore, the acceleration is not constant, nor is it zero; it is increasing.
Final Solution: The acceleration of the object is increasing, as the velocity is changing at a non-uniform rate indicated by the cubic relationship between distance and time.
More Questions from Class 9 Science Force and Laws of Motion Additional Exercises
- QA1(b): The following is the distance-time table of an object in motion: (b) What do you infer about the forces acting on the object?
- QA2: Two persons manage to push a motorcar of mass 1200 kg at a uniform velocity along a level road. The same motorcar can be pushed by three persons to produce an acceleration of 0.2 $m s^{-2}$. With what force does each person push the motorcar? (Assume that all persons push the motorcar with the same muscular effort.)
- QA3: A hammer of mass 500 g, moving at 50 $m s^{-1}$, strikes a nail. The nail stops the hammer in a very short time of 0.01 s. What is the force of the nail on the hammer?
- QA4: A motorcar of mass 1200 kg is moving along a straight line with a uniform velocity of 90 km/h. Its velocity is slowed down to 18 km/h in 4 s by an unbalanced external force. Calculate the acceleration and change in momentum. Also calculate the magnitude of the force required.
CBSE Solutions for Class 9 Science Force and Laws of Motion
Chapters in CBSE - Class 9 Science
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