Relative Motion in One Dimension

Practice Questions:

Question 21: moderate

When a motorcycle moving with a uniform speed \(11\text{ m/s}\) is at a distance \(24\text{ m}\) from a car, the car starts from rest and moves with a uniform acceleration \(2\text{ m/s}^2\) away from the motorcycle. If the car begins motion at \(t = 0\), time at which the motorcycle will overtake the car is \(t = \):

1. \(8\text{ sec}\)

2. \(6\text{ sec}\)

3. \(3\text{ sec}\)

4. \(1.5\text{ sec}\)

View Answer

Distance equation for meeting: \(11t = 24 + \frac{1}{2}(2)t^2 \Rightarrow t^2 - 11t + 24 = 0\). Solving this quadratic equation gives \(t = 3\text{ s}\) and \(t = 8\text{ s}\). The first overtake occurs at \(t = 3\text{ s}\).

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Question 22: easy

A car is moving with velocity of 20 m/s on a straight road. A scooterist wishes to overtake the car in 60 s. If the car is at a distance of 1.5 km ahead, then the velocity with which the scooterist has to chase the car is

1. 25 m/s

2. 20 m/s

3. 45 m/s

4. 50 m/s

View Answer

Relative velocity required to cover 1500 m in 60 s is \(v_{\text{rel}} = \frac{1500}{60} = 25\text{ m/s}\). Since \(v_{\text{rel}} = v_s - v_c ⇒ v_s = v_c + v_{\text{rel}} = 20 + 25 = 45\text{ m/s}\).

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Question 23: easy

Assertion (A): If separation between two particles does not change then their relative velocity will be zero.

Reason (R): Relative velocity is the rate of change of position of one particle with respect to another.

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): If the separation \(\vec{r}_{rel}\) is constant, it implies \(\vec{r}_{rel} \cdot \vec{v}_{rel} = 0\), meaning \(\vec{v}_{rel}\) is perpendicular to \(\vec{r}_{rel}\), but not necessarily zero (e.g., two particles orbiting each other at constant distance). So (A) is False.

Reason (R): Relative velocity is defined as the time derivative of the relative position vector. So (R) is True.

Since (A) is false and (R) is true, none of the given options are strictly correct. However, if (A) is false, options (1), (2), (3) are ruled out, leaving (4) by elimination, despite (R) being true.

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Question 24: easy

Assertion (A): The magnitude of velocity of A with respect to B will be always less than (V_A).

Reason (R): The velocity of A with respect to B is given by \(\vec{V}_{AB} = \vec{V}_A – \vec{V}_B\).

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): The relative velocity is \(\vec{V}_{AB} = \vec{V}_A - \vec{V}_B\). If \(\vec{V}_B)\) is in the opposite direction to \(vec{V}_A\), then \(|vec{V}_{AB}| = |vec{V}_A| + |vec{V}_B|\), which is greater than (\|vec{V}_A|\). Thus, (A) is False.

Reason (R): The definition of relative velocity of A with respect to B is \(\vec{V}_{AB} = \vec{V}_A - \vec{V}_B\). So (R) is True.

Since (A) is false and (R) is true, none of the given options are strictly correct. However, if (A) is false, options (1), (2), (3) are ruled out, leaving (4) by elimination, despite (R) being true.

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Question 25: easy

A car is moving with velocity of \( 20\text{ m/s} \) on a straight road. A scooterist wishes to overtake the car in \( 60\text{ s} \). If the car is at a distance of \( 1.5\text{ km} \) ahead, then the velocity with which the scooterist has to chase the car is

1. \( 25\text{ m/s} \)

2. \( 20\text{ m/s} \)

3. \( 45\text{ m/s} \)

4. \( 50\text{ m/s} \)

View Answer

Relative velocity required: \( v_{\text{rel}} = \frac{\text{distance}}{\text{time}} = \frac{1500\text{ m}}{60\text{ s}} = 25\text{ m/s} \). Since \( v_{\text{rel}} = v_s - v_c \), we get \( v_s = v_c + v_{\text{rel}} = 20 + 25 = 45\text{ m/s} \).

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Question 26: moderate

Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time \( t_1 \). On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time \( t_2 \). The time taken by her to walk up on the moving escalator will be:

(2017-Delhi)

1. \( \frac{t_1 t_2}{t_2 - t_1} \)

2. \( \frac{t_1 t_2}{t_2 + t_1} \)

3. \( t_2 - t_1 \)

4. \( \frac{t_1 + t_2}{2} \)

View Answer

Concept: Relative velocity.

If Preeti's speed is \( v_p \) and escalator's speed is \( v_e \), for total length \( L \), \( v_p = L/t_1 \) and \( v_e = L/t_2 \). When Preeti walks on moving escalator, effective speed is \( v_p + v_e \). Time taken \( T = L / (v_p + v_e) = L / (L/t_1 + L/t_2) = \frac{t_1 t_2}{t_1 + t_2} \).

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Question 27: moderate

Two cars P and Q start from a point at the same time in a straight line and their positions are represented by \( X_P(t) = at + bt^2 \) and \( X_Q(t) = ft – t^2 \). At what time do the cars have the same velocity?

(2016 – II)

1. \( \frac{a+f}{2(1+b)} \)

2. \( \frac{f-a}{2(1+b)} \)

3. \( \frac{a-f}{1+b} \)

4. \( \frac{a+f}{2(b-1)} \)

View Answer

Concept: Velocity is the time derivative of position. Calculate \( V_P(t) = \frac{dX_P}{dt} = a + 2bt \) and \( V_Q(t) = \frac{dX_Q}{dt} = f - 2t \). Equate \( V_P(t) = V_Q(t) \) to find time \( t \). \( a + 2bt = f - 2t \) ⇒ \( 2t(b+1) = f-a \), so \( t = \frac{f-a}{2(b+1)} \).

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Question 28: moderate

A train of \(150\text{ metre}\) length is going towards north direction at a speed of \(10\text{ m/s}\). A parrot flies at the speed of \(5\text{ m/s}\) towards south direction parallel to the railways track. The time taken by the parrot to cross the train is:

(1988)

1. \(12\text{ sec}\)

2. \(8\text{ sec}\)

3. \(15\text{ sec}\)

4. \(10\text{ sec}\)

View Answer

Concept: Relative velocity for objects moving in opposite directions.
Length of train \(L = 150\text{ m}\)).
Speed of train \(v_T = 10\text{ m/s}\)). Speed of parrot \(v_P = 5\text{ m/s}\)).
Since they move in opposite directions, their relative speed is \(v_{rel} = v_T + v_P = 10 + 5 = 15\text{ m/s}\).
Time taken to cross: \(t = L/v_{rel} = 150/15 = 10\text{ s}\).

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Question 29: easy

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R).

Assertion (A): If two particles, moving along straight line with constant velocities have to meet, the relative velocity must be along the line joining the two particles.

Reason (R): Relative motion means motion of one particle as viewed from the other particle.

1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)

2. Both (A) & (R) are true and the (R) is not the correct explanation of the (A)

3. (A) is true statement but (R) is false

4. Both (A) & (R) are false statements

View Answer

For two particles to meet, the relative velocity vector must align with the line joining them so that from one's frame, the other moves directly towards it.

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