Question

A stone is dropped from a height of 45 m. What is the time taken for the stone to reach the ground?

• A. 3 s
• B. 5 s
• C. 6 s
• D. 9 s

Hint:

This is a case of free fall starting from rest, so the initial velocity is zero and acceleration equals g (take $g = 10\,m/s^2$ unless a different value is given). You can either apply $h = \frac{1}{2}gt^2$ directly, or first find the final velocity using $v^2 = 2gh$ and then use the average velocity to reach the time, both routes work equally well.

Answer 1

The Correct Option is A

The problem involves calculating the time taken for an object to fall from a certain height under gravity. Here, we use the equation of motion for free fall without initial velocity:
h = ¹⁄₂ gt²

Where:

• h = height (45 m)
• g = acceleration due to gravity (9.8 m/s²)
• t = time in seconds

Substitute the values into the equation:
45 m = ¹⁄₂ (9.8 m/s²) t²
Solving for t²:
45 m = 4.9t²
t² = ^{45 m}⁄_{4.9 m/s²}
t² = 9.18
t ≈ √9.18 ≈ 3.03 s

Rounding to the nearest second, t ≈ 3 s.

Therefore, the time taken for the stone to reach the ground is approximately 3 seconds, confirming the correct answer is 3 s.

Answer 2

Concept:

• For an object released from rest and falling freely, the equation $v^2 = u^2 + 2gh$ gives the final velocity directly, without needing time.
• For uniformly accelerated motion starting from rest, the average velocity equals half the final velocity, since velocity increases uniformly from $0$ to $v$.
• Distance covered equals average velocity multiplied by time, so time can be obtained once the average velocity is known.

Step 1: Find the final velocity using $v^2 = u^2 + 2gh$.
Taking $u = 0$, $h = 45\,m$, and $g = 10\,m/s^2$ (the standard value used in such problems),
$v^2 = 0 + 2(10)(45) = 900$
$v = 30\,m/s$

Step 2: Find the average velocity during the fall.
Since the motion starts from rest and acceleration is uniform,
$v_{avg} = \dfrac{u+v}{2} = \dfrac{0+30}{2} = 15\,m/s$

Step 3: Use distance equals average velocity multiplied by time to find the time taken.
$h = v_{avg} \times t$
$45 = 15 \times t$
$t = 3\,s$

Final Answer: $t = 3$ seconds