Math Problem Statement
Solution
Let's break down the problem for parts (a), (b), and (c):
Problem Analysis:
-
Tortoise:
- Speed: (constant)
- Distance to cover: 60 meters
- Time for the tortoise to finish the race:
-
Hare:
- The hare starts 100 seconds later.
- Accelerates uniformly for 10 seconds, reaching a top speed of .
- This means the acceleration is given by:
- Distance covered during acceleration:
- Maintains top speed for 10 seconds:
- Distance during this period:
- Decelerates to rest in 10 seconds:
- Deceleration:
- Distance during deceleration:
- Total distance covered by the hare:
- Time taken for the hare to finish the race:
(a) Plotting the Velocity-Time Graph:
-
Tortoise:
- Constant velocity of for 120 seconds.
-
Hare:
- Accelerates uniformly from 0 to over 10 seconds.
- Maintains speed of for 10 seconds.
- Decelerates uniformly back to 0 over 10 seconds.
- Starts 100 seconds after the tortoise.
(b) Who Wins the Race?
- The tortoise takes 120 seconds to finish.
- The hare starts after 100 seconds and finishes in 30 seconds. Thus, the hare finishes at .
Therefore, the tortoise wins, finishing 10 seconds earlier than the hare.
(c) Time Between Crossings:
- The tortoise finishes in 120 seconds.
- The hare finishes in 130 seconds.
So, the time between the two animals crossing the finish line is:
Would you like me to sketch the velocity-time graph for you? Do you have any questions about the calculations?
Here are 5 related questions you might consider:
- How does the total distance traveled change if the hare starts accelerating sooner?
- What would happen if the tortoise had a speed boost midway through the race?
- If the race was 100 meters long instead, how would the results change?
- What would happen if the hare's acceleration was faster or slower?
- How can the time of the hare be minimized to finish the race faster?
Tip: Always compare the areas under velocity-time graphs when finding distances traveled by different objects.
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Math Problem Analysis
Mathematical Concepts
Kinematics
Uniform Acceleration
Velocity-Time Graphs
Formulas
v = u + at (Final velocity)
s = ut + 1/2 at^2 (Displacement during acceleration)
s = vt (Displacement at constant velocity)
Area under velocity-time graph = displacement
Theorems
Constant Speed Theorem
Uniform Acceleration Theorem
Suitable Grade Level
Grades 9-12