Two objects A and B are projected up with velocities 20 ms^–1 and 30 ms^–1 respectively. The ratio of the distances travelled by them in the last second of their ascending motion is

The Correct answer is 1 : 1.

Key Points

• When an object is projected upwards, it decelerates due to the effect of gravity.
• At the peak of its motion, its velocity becomes zero. The time taken to reach this point is determined by the initial velocity and the acceleration due to gravity (g = 9.8 m/s²).
• The distance covered by the object in the last second of its ascending motion is the same for any object regardless of its initial velocity. This is because, during the last second, the velocity decreases linearly due to constant deceleration (gravity).
• For object A, initial velocity = 20 m/s, and for object B, initial velocity = 30 m/s. However, the distance covered in the last second of ascending motion depends only on the uniform deceleration and not on the initial velocity.
• Using the equations of motion and analyzing the distances covered in the last second, it can be mathematically proven that the ratio of the distances covered is 1 : 1.
• This phenomenon is independent of the difference in initial velocities because the deceleration due to gravity is the same for both objects.

Additional Information

• The equation of motion \( v = u - g\times t\) is used to calculate the velocity at any instant of time. Here, v is the final velocity, u is the initial velocity, g is the acceleration due to gravity, and t is the time.
• The distance covered in the nth second of motion can be calculated using the formula: \(S_n = u + \frac{1}{2}a(2n - 1)\), where a is the acceleration (here, -g).
• In the last second of upward motion, the object’s velocity will reduce to zero, and the distance covered will be the same irrespective of the initial velocity.