Vertical Motion Distance Ratio
Question: A person throws a ball vertically upwards from level ground with certain velocity such that it comes back to the ground in time \( t \). The distance traveled in the first \( \frac{t}{3} \) time is \( s_1 \), and the distance traveled in the next \( \frac{t}{3} \) time is \( s_2 \). The ratio \( s_2 : s_1 \) is:
Detailed Explanation
When a ball is thrown vertically upwards, it follows a symmetric parabolic path under constant gravitational acceleration \( g \). The total time of flight is \( t \), meaning it takes \( \frac{t}{2} \) to reach the maximum height and another \( \frac{t}{2} \) to return to the ground.
Step 1: Initial Velocity Calculation
Using the kinematic equation for displacement:
\[
y(t) = u t - \frac{1}{2} g t^2
\]
At \( t \), \( y(t) = 0 \):
\[
0 = u t - \frac{1}{2} g t^2 \implies u = \frac{g t}{2}
\]
Step 2: Positions at Key Times
- At \( t = 0 \): \( y(0) = 0 \)
- At \( t = \frac{t}{3} \): \( y\left(\frac{t}{3}\right) = \frac{g t^2}{9} \)
- At \( t = \frac{t}{2} \): \( y\left(\frac{t}{2}\right) = \frac{g t^2}{8} \) (maximum height)
- At \( t = \frac{2t}{3} \): \( y\left(\frac{2t}{3}\right) = \frac{g t^2}{9} \)
Step 3: Distance Traveled in Each Interval
- \( s_1 \) (0 to \( \frac{t}{3} \)): \( s_1 = y\left(\frac{t}{3}\right) - y(0) = \frac{g t^2}{9} \)
- \( s_2 \) (\( \frac{t}{3} \) to \( \frac{2t}{3} \)): During this interval, the ball goes up to the peak and then comes down to the same height.
Distance up: \( y\left(\frac{t}{2}\right) - y\left(\frac{t}{3}\right) = \frac{g t^2}{8} - \frac{g t^2}{9} = \frac{g t^2}{72} \)
Distance down: \( y\left(\frac{t}{2}\right) - y\left(\frac{2t}{3}\right) = \frac{g t^2}{8} - \frac{g t^2}{9} = \frac{g t^2}{72} \)
Total \( s_2 = \frac{g t^2}{72} + \frac{g t^2}{72} = \frac{g t^2}{36} \)
Step 4: Compute the Ratio
\[
\frac{s_2}{s_1} = \frac{\frac{g t^2}{36}}{\frac{g t^2}{9}} = \frac{1}{36} \times \frac{9}{1} = \frac{1}{4}
\]
Thus, \( s_2 : s_1 = 1 : 4 \).
Final Answer: The ratio \( s_2 : s_1 \) is \( 1 : 4 \), which corresponds to option (3).
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