The Problem:
If a ball is placed on a straight sloping track and then released from rest, the distances that it moves in successive equal intervals of time are found to be in the ratio 1:3:5:7:…. Show that this is consistent with the theory that the ball rolls down the track with constant acceleration.
HandWritten solution
Ball Rolling Down a Sloping Track
Ball Rolling Down a Sloping Track
Given Data:
- The ball is released from rest.
- It moves in successive equal time intervals with distances in the ratio 1:3:5:7…
- We need to show this is consistent with constant acceleration.
Step 1: Distance Equation for Constant Acceleration
The formula for distance traveled with constant acceleration is:
s = ut + (1/2) a t²
Since the ball starts from rest, u = 0, so:
s = (1/2) a t²
Step 2: Find Distances in Successive Time Intervals
Let T be the time interval.
First Interval (0 to T):
s1 = (1/2) a T²
Second Interval (0 to 2T):
s2 = (1/2) a (2T)² = 2 a T²
Third Interval (0 to 3T):
s3 = (1/2) a (3T)² = 4.5 a T²
Step 3: Find Individual Distances in Each Interval
The distance traveled in each interval is:
First Interval:
d1 = s1 = (1/2) a T²
Second Interval:
d2 = s2 – s1 = 2 a T² – (1/2) a T² = (3/2) a T²
Third Interval:
d3 = s3 – s2 = 4.5 a T² – 2 a T² = (5/2) a T²
Fourth Interval:
d4 = s4 – s3 = 8 a T² – 4.5 a T² = (7/2) a T²
Step 4: Verify the Ratio
The distances follow:
1 : 3 : 5 : 7
Final Conclusion
Since the distances follow this pattern, the ball moves with constant acceleration. ✅
