Assertion (A): When a particle is observed from two different inertial reference frames the general shape of the trajectory of particle is same.
Reason (R): The position vector of a particle and its velocity are frame independent quantities.
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
The general shape of a particle's trajectory is invariant across inertial reference frames. However, position vectors and velocities are frame-dependent quantities, changing with the relative motion of frames. Therefore, assertion (A) is true, but reason (R) is false.
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): Two bodies of masses \(M\) and \(m\) (\(M > m\)) are allowed to fall from the same height if the air resistance force for each be the same then both the bodies will reach the earth simultaneously.
Reason (R): For same air resistance, acceleration of both the bodies will be same.
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
If the air resistance force \(F_{/text{air}}\) is the same for both bodies, the net force is \(mg - F_{/text{air}}\). The acceleration is \(a = g - F_{/text{air}}/m\). Since masses \(M\) and \(m\) are different, their accelerations will be different. Thus, they will not reach the earth simultaneously, and their accelerations will not be the same. Both assertion and reason are false.
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): Displacement of a body is vector sum of the area under velocity-time graph.
Reason (R): Displacement is a vector quantity.
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
The area under a velocity-time graph indeed represents displacement, considering areas above the time axis as positive and below as negative, which is a vector sum.
Displacement is a vector quantity, meaning it has both magnitude and direction. This vector nature directly explains why the signed area (vector sum) under the velocity-time graph yields displacement.
Both (A) and (R) are true, and (R) correctly explains (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): For a moving particle on a straight line magnitude of average velocity between any two points will be less than magnitude of instantaneous velocity at every point between them.
Reason (R): In \(x-t\) graph slope of chord joining two points gives average velocity between them.
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. The magnitude of average velocity can be equal to or greater than the magnitude of instantaneous velocity at some points, or less than at others. For constant velocity, they are equal.
Reason (R) is true. By definition, the slope of the chord in an \(x-t\) graph represents the average velocity. However, since Assertion (A) is false, option (4) is selected.
Assertion (A): If initial velocity is negative but acceleration is positive then displacement of a particle can never be positive.
Reason (R): If initial velocity is negative and acceleration is positive then motion must be retarded throughout.
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. If initial velocity is negative and acceleration is positive, the particle can eventually move in the positive direction, leading to a positive displacement (e.g., \(s = v_0 t + \frac{1}{2}at^2\) can be positive for large \(t\)).
Reason (R) is false. Motion is initially retarded, but as velocity becomes positive (due to positive acceleration), the motion becomes accelerated.
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): 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.