Assertion (A): The speedometer of an automobile measures the average speed of the automobile.
Reason (R): Average velocity is equal to total distance divided by total time taken.
1. (1) Both (A) \& (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) \& (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
View Answer
A speedometer measures instantaneous speed, not average speed. Average velocity is defined as total displacement divided by total time, while average speed is total distance divided by total time. Both assertion (A) and reason (R) are incorrect.
Assertion (A): The average speed of an object may be equal to arithmetic mean of individual speeds.
Reason (R): The average speed is equal to total distance travelled per total time taken.
1. (1) Both (A) \& (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) \& (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
View Answer
Average speed is defined as total distance divided by total time, making reason (R) true.
Assertion (A) is also true, as average speed can be equal to the arithmetic mean of individual speeds if the time intervals for those speeds are equal. However, reason (R) only defines average speed, it does not explain the condition under which it equals the arithmetic mean.
Assertion (A): \(|\Delta v| / \Delta t\) and \(\Delta |v| / \Delta t\) are same if particle is moving in one dimension.
Reason (R): In one dimensional motion there is no component of acceleration perpendicular to velocity.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is true if the particle does not reverse its direction of motion; otherwise, it is generally false. Assuming this condition for 'moving in one dimension' for the purpose of the question.
Reason (R) is true; in one dimension, velocity and acceleration are collinear.
R is not a correct explanation for A, as the absence of perpendicular acceleration components does not directly imply the equality of magnitude of average acceleration and average rate of change of speed when velocity changes direction.
Thus, both A and R are true, but R is not the correct explanation of A.
Assertion (A): If velocity of a particle moving in a straight line is zero at a point, its acceleration will be zero at that point.
Reason (R): Wherever \(a = v \frac{dv}{dx}\) holds, \(v = 0 \Rightarrow a = 0\).
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is false. For example, a ball thrown vertically upwards has zero velocity at its highest point, but its acceleration is \(g\).
Reason (R) is false. While the formula \(a = v \frac{dv}{dx}\) is correct, the implication \(v = 0 \Rightarrow a = 0\) from this formula is physically incorrect as \(dv/dx\) itself might be undefined or lead to physically inconsistent conclusions when \(v=0\). Physically, \(a = dv/dt\), which can be non-zero when \(v=0\).
Assertion (A): A body is thrown vertically upwards with an initial speed \( 25 \text{ m/s} \) from a position 1. It falls back to position 1 after some time. During this time duration, total change of velocity of the body is zero.
Reason (R): Average acceleration of the body during this time is zero.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
View Answer
Assertion (A) is false. Initial velocity is \( +25 \text{ m/s} \). Final velocity at the same position is \( -25 \text{ m/s} \). The change in velocity is \( \Delta \vec{v} = (-25) - (+25) = -50 \text{ m/s} \). Reason (R) is false. Since the change in velocity \( \Delta vec{v} \) is not zero, and \( \Delta t \) is a finite time, the average acceleration \(\ vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t} \) is also not zero. It is \( -g \).
Therefore, both the Assertion and the Reason are false.
Assertion (A): A particle moves in a straight line with constant acceleration. The average velocity of this particle can not be zero in any time interval.
Reason (R): For a particle moving in straight line, the average velocity in a time interval is always ((frac{u+v}{2})), where (u) and (v) are initial and final velocities of the particle in given time interval.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Solution: (A) is false; average velocity can be zero if displacement is zero (e.g., object returns to start with constant acceleration). (R) is false; the formula \(v_{avg} = \frac{u+v}{2}\) is only valid for constant acceleration, not 'always' for any straight line motion.
Assertion (A): At any instant, acceleration of a body can change its direction without any change in the direction of velocity.
Reason (R): At any instant, direction of acceleration is same as that of direction of change in velocity vector at that instant.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Concept: Relationship between acceleration and velocity.
Formula: \(vec{a} = \frac{d\vec{v}}{dt}\).
Solution: (A) is true. For example, a car moving straight can accelerate forward, then brake (accelerate backward) while its velocity direction remains forward. (R) is true; acceleration is defined as the rate of change of velocity, so its direction is the same as the direction of the change in velocity. (R) correctly explains (A) by providing the fundamental definition.