Numerical Problems Based on Position-Time and Velocity-Time Graphs for Class 11 Physics

(i) Highest and lowest speed:
In a position-time graph, the slope of the line represents the speed. The line for car C is the steepest, so C has the highest speed. The line for car A is the least steep, so A has the lowest speed.

(ii) Meeting at the same point:
For all three cars to be at the same point simultaneously, all three lines must intersect at a single common point. Since they do not, they are never at the same point on the road.

(iii) Position of B when A passes C:
Locate the intersection point of lines A and C. Looking at the corresponding time on the x-axis, trace up to line B and read its position on the y-axis, which corresponds to $6\text{ km}$ from the origin.

(iv) Distance A travels between passing B and C:
Find the position where A intersects B, and the position where A intersects C. The difference between these two position values on the y-axis is $3\text{ km}$.

(v) Relative velocity of C with respect to A:
Using the slopes from the standard graph, $v_C \approx 10\text{ kmh}^{-1}$ and $v_A \approx 3\text{ kmh}^{-1}$.
Relative velocity $v_{CA} = v_C – v_A = 10 – 3 = 7\text{ kmh}^{-1}$.

(vi) Relative velocity of B with respect to C:
Using the slopes, $v_B \approx 8\text{ kmh}^{-1}$ and $v_C \approx 10\text{ kmh}^{-1}$.
Relative velocity $v_{BC} = v_B – v_C = 8 – 10 = -2\text{ kmh}^{-1}$.

The insect crawls up $5\text{ cm}$ and slides down $3\text{ cm}$ over a period of 2 minutes.

Net distance covered in 2 minutes = $5 – 3 = 2\text{ cm}$.

We must be careful because in the final minute, the insect will reach the crevice before having a chance to slide down. Let’s find the time taken to reach a height just below the final $5\text{ cm}$ push ($24 – 5 = 19\text{ cm}$).

To reach $18\text{ cm}$ (which is a multiple of $2\text{ cm}$), the insect takes:

At $t = 18\text{ min}$, the insect is at $18\text{ cm}$.
At $t = 19\text{ min}$, it crawls up $5\text{ cm}$ to reach $18 + 5 = 23\text{ cm}$.
At $t = 20\text{ min}$, it slides down $3\text{ cm}$ to reach $23 – 3 = 20\text{ cm}$.
At $t = 21\text{ min}$, it crawls up $5\text{ cm}$ to reach $20 + 5 = 25\text{ cm}$.

Since $25\text{ cm}$ is greater than $24\text{ cm}$, the insect reaches the crevice during the 21st minute.

Graph description: The position-time graph will be a zigzag line (sawtooth pattern), with steeper positive slopes for the odd minutes (crawling up) and less steep negative slopes for the even minutes (sliding down).

First, convert speeds from $\text{kmh}^{-1}$ to $\text{ms}^{-1}$.

First Car:
Initial velocity, $u_1 = 52 \times \frac{5}{18} = 14.44\text{ ms}^{-1}$.
Time, $t_1 = 5\text{ s}$. Final velocity, $v_1 = 0$.
Distance travelled is the area under the speed-time graph (a triangle):

Second Car:
Initial velocity, $u_2 = 34 \times \frac{5}{18} = 9.44\text{ ms}^{-1}$.
Time, $t_2 = 10\text{ s}$. Final velocity, $v_2 = 0$.
Distance travelled:

Since $S_2 > S_1$ ($47.2\text{ m} > 36.1\text{ m}$), the second car travelled farther.

(i) Acceleration (Stage 1):
Starts from rest ($u_1 = 0$), reaches $v_1 = 8\text{ ms}^{-1}$ in $t_1 = 8\text{ s}$.

Distance covered in Stage 1 ($S_1$):

(ii) Retardation (Stage 3):
Initial velocity $u_3 = 8\text{ ms}^{-1}$, final velocity $v_3 = 0$, distance $S_3 = 32\text{ m}$.

The retardation is $1\text{ ms}^{-2}$.
Time taken for Stage 3 ($t_3$):

(iii) Total Time (Stage 2 included):
Total distance $S_{total} = 464\text{ m}$. Distance for Stage 2 (uniform velocity):

Since it moves with uniform velocity $v_2 = 8\text{ ms}^{-1}$:

Total time taken:

The motion is divided into two parts.

Part 1: Uniform Acceleration ($0$ to $5\text{ s}$)
Initial velocity $u = 0$, acceleration $a = 3\text{ ms}^{-2}$, time $t_1 = 5\text{ s}$.

Velocity attained at the end of 5 seconds ($v$):

Distance covered in the first 5 seconds ($S_1$):

Part 2: Uniform Velocity ($5\text{ s}$ to $7\text{ s}$)
The car moves with a constant velocity of $15\text{ ms}^{-1}$ for the remaining $t_2 = 7 – 5 = 2\text{ s}$.

Distance covered in these 2 seconds ($S_2$):

Total distance:

(Note: In a distance-time graph, the curve would be a parabola opening upwards from $t=0$ to $t=5$, and then a straight line with a constant positive slope from $t=5$ to $t=7$).