✓ Correct Answer: C — 15 m/s
How to analyse the problem: The acceleration changes after the first time interval. Therefore, the motion must be treated as two separate uniformly accelerated stages.
How to crack it: Solve each time interval separately. The final velocity of the first interval becomes the initial velocity of the second interval. Find the distance covered in each interval, add them, and divide by the total time.
Basic formula:
For multiple intervals:
Total time:
Therefore:
Conclusion: When acceleration changes from one time interval to another, calculate each stage separately and use total distance divided by total time. Do not simply average the accelerations.
First interval:
Distance covered in the first interval:
Speed at the end of the first interval:
Second interval:
Distance covered in the second interval:
Total distance:
Total time:
Therefore:
✓ Correct Answer: C — 15 m/s
How to analyse the problem: The acceleration changes after a specified distance rather than after a specified time. Therefore, the motion must be divided into separate distance intervals.
How to crack it: For each distance interval, find the time taken using the appropriate equation of motion. The final velocity of one interval becomes the initial velocity of the next interval. Finally, add all the times and use total distance divided by total time.
Basic formulas:
For each distance interval:
Total time is:
Therefore:
Conclusion: When acceleration changes after different distances, solve each distance interval separately. The average speed is always obtained from total distance divided by total time.
First distance interval:
Find the velocity at the end of the first interval:
Now find the time for the first interval:
Second distance interval:
Find the final velocity:
Find the time for the second interval:
Total distance:
Total time:
Therefore:
✓ Correct Answer: B — 15 m/s
How to analyse the problem: The motion has two different stages. In the first stage, the car accelerates uniformly. In the second stage, it moves with uniform speed.
How to crack it: Treat each stage separately. Find the distance covered during the accelerated stage and then the distance covered during the uniform speed stage. Add the distances and divide by the total time.
Basic formulas:
For the uniform-speed stage:
Finally:
Conclusion: When a journey contains both uniformly accelerated motion and uniform-speed motion, solve each stage separately and use total distance divided by total time.
Stage 1: Uniform acceleration
Distance covered during acceleration:
Speed reached at the end of Stage 1:
Stage 2: Uniform speed
The car continues at 20 m/s for another 5 s.
Total distance:
Total time:
Therefore:
✓ Correct Answer: B — 4 m/s
How to analyse the problem: The car moves along a complete circular path. Therefore, the distance travelled is equal to the circumference of the circle.
How to crack it: First find the circumference of the circle. This gives the total distance travelled in one complete revolution. Then divide the total distance by the total time.
Basic formula:
For one complete revolution, the distance travelled is the circumference:
Therefore, for one complete revolution:
Conclusion: For one complete revolution, use the circumference 2πr as the total distance and divide it by the total time.
Step 1: Find the distance travelled
The car completes one complete revolution. Therefore, the distance travelled is the circumference of the circle.
Step 2: Find the average speed
Total time taken is 11 s.
✓ Correct Answer: B — 2 m/s
How to analyse the problem: The particle moves along a semicircular path. A semicircle is half of a complete circular path, so its path length is half of the circumference.
How to crack it: Find the length of the semicircular path first. Then use total distance divided by total time to calculate average speed.
Basic formula:
The circumference of a complete circle is:
Since a semicircle is half of a complete circle:
Therefore:
Conclusion: For motion along a semicircular path, the actual distance travelled is πr. Use this curved distance when calculating average speed.
Step 1: Find the distance travelled
The particle travels along a semicircle. Therefore, the distance travelled is:
Step 2: Find the average speed
Total time taken is 22 s.
✓ Correct Answer: B — 2 m/s
How to analyse the problem: The particle moves through one-fourth of a complete circular path. Therefore, the distance travelled is one-fourth of the circumference of the circle.
How to crack it: First find the circumference of the complete circle. Then take one-fourth of it to obtain the actual distance travelled along the quarter-circular path. Finally, divide the distance by the total time.
Basic formula:
Circumference of a complete circle:
A quarter circle is one-fourth of the complete circumference:
Therefore:
Conclusion: For a quarter-circular path, the actual distance travelled is πr/2. Average speed is obtained using this curved distance and the total time.
Step 1: Find the distance travelled
The particle travels through one-fourth of a complete circle. Therefore:
Step 2: Find the average speed
Total time taken is 11 s.
✓ Correct Answer: B — 4 m/s
How to analyse the problem: The particle completes more than one revolution around the circular track. Therefore, the total distance is the distance covered in one revolution multiplied by the number of revolutions.
How to crack it: First find the circumference of the circular track. Then multiply it by the total number of revolutions to obtain the total distance. Finally, divide the total distance by the total time.
Basic formula:
Distance covered in one complete revolution:
If the particle completes n revolutions, the total distance is:
Therefore:
Conclusion: For multiple complete revolutions, multiply the circumference by the number of revolutions to obtain the total distance.
Step 1: Find the distance covered in one revolution
Step 2: Find the total distance
The particle completes 5 revolutions.
Step 3: Find the average speed
Total time taken is 55 s.
✓ Correct Answer: B — 5.6 m/s
How to analyse the problem: The motion consists of two different parts. First, the particle travels along a complete circular path. Then it travels along a straight path. Therefore, the total distance is the sum of the distances covered in both parts.
How to crack it: Calculate the circular distance using the circumference formula. Then add the straight-line distance. Finally, divide the total distance by the total time.
Basic formulas:
Average speed is:
Conclusion: When a journey contains a circular part and a straight-line part, calculate the actual distance of each part separately, add them, and divide by the total time.
Step 1: Find the circular distance
The particle completes one complete revolution.
Step 2: Find the total distance
The straight-line distance is 12 m.
Step 3: Find the average speed
Total time taken is 10 s.
✓ Correct Answer: B — 4 m/s, 0 m/s
How to analyse the problem: The particle completes one complete revolution and returns to its starting point. Therefore, the distance travelled is the complete circumference, while the displacement is zero.
How to crack it: For average speed, use the total distance travelled. For average velocity, use the displacement. Do not use displacement to calculate average speed.
Basic formulas:
For one complete revolution:
Since the particle returns to its starting point:
Conclusion: After one complete revolution, the particle has travelled a non-zero distance but its displacement is zero. Therefore, average speed is non-zero, while average velocity is zero.
Step 1: Find the total distance
The particle completes one complete revolution. Therefore, the distance travelled is the circumference of the circle.
Step 2: Find the average speed
Total time taken is 22 s.
Step 3: Find the displacement
After one complete revolution, the particle returns to its initial position. Hence, its displacement is zero.
Therefore, the magnitude of average velocity is:
✓ Correct Answer: C — 4 m/s
How to analyse the problem: The particle travels along all three sides of a triangle. Therefore, the total distance is the sum of the lengths of all three sides.
How to crack it: Add the distances travelled along the three sides to obtain the total distance. Then add the time taken for each side to obtain the total time. Finally, use total distance divided by total time.
Basic formula:
For motion along three sides:
The total time is:
Conclusion: When a particle moves along the sides of a triangle, calculate the actual distance covered along each side and add all the distances before calculating average speed.
Step 1: Find the total distance
The triangle is equilateral, so all three sides have the same length.
Step 2: Find the total time
The particle takes 5 s along each side.
Step 3: Find the average speed