An object is thrown vertically upwards with a velocity of 120 m/s. After 0.75 seconds another object of mass 500 gm is thrown in the same manner with a velocity of 150 m/s. Prove mathematically which object will hit the ground first.

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We have a projectile motion problem involving two objects thrown vertically upward. The first object is thrown with an initial velocity of 120 meters per second. After 0.75 seconds, a second object with mass 500 grams is thrown upward with an initial velocity of 150 meters per second. We need to determine which object will hit the ground first. Let's analyze the first object. We use the equation of motion s equals u t plus one half a t squared. Since the object returns to ground level, the displacement s is zero. The initial velocity u is 120 meters per second, and acceleration a is negative 9.8 meters per second squared due to gravity. Substituting these values, we get zero equals 120 t plus one half times negative 9.8 times t squared. Factoring out t, we get t times 120 minus 4.9 t equals zero. The non-zero solution gives us t equals 120 divided by 4.9, which equals 24.49 seconds. Now let's analyze the second object. We use the same equation of motion. For object 2, the initial velocity is 150 meters per second. Setting displacement to zero for when it returns to ground level, we get zero equals 150 t flight plus one half times negative 9.8 times t flight squared. Factoring out t flight, we get t flight times 150 minus 4.9 t flight equals zero. The non-zero solution gives us t flight equals 150 divided by 4.9, which equals 30.61 seconds. This is the flight time of object 2 after it is launched. Now we calculate the total time for object 2. Since object 2 is launched 0.75 seconds after object 1, its total time is the delay plus its flight time. T2 equals 0.75 plus 30.61, which equals 31.36 seconds. Comparing the total times: object 1 takes 24.49 seconds, while object 2 takes 31.36 seconds. Since T1 is less than T2, object 1 hits the ground first. Note that the mass of object 2 does not affect this calculation, as all objects fall at the same rate in the absence of air resistance. To summarize our analysis: Object 1 hits the ground first at 24.49 seconds, while Object 2 takes 31.36 seconds. The 0.75 second launch delay significantly affects the total time. Importantly, the mass of the objects does not influence the timing in projectile motion. Our mathematical analysis using kinematic equations proves this result conclusively.