A person is thrown upward from a height of 2 meters above Earth’s surface with an initial velocity of 50 m/s, and the only force acting on them is gravity (9.8 m/s2 ).
The correct answer and explanation is:
Okay, the question implies a need to analyze the motion of the person under gravity. The most common calculation requested in such a scenario is the time it takes for the person to reach a specific point, such as their maximum height or, more dramatically, the ground. Let’s calculate the time it takes for the person to hit the ground (y=0).
Correct Answer:
The positive time it takes for the person to hit the ground is approximately 10.24 seconds.
Explanation:
This problem involves one-dimensional motion under constant acceleration due to gravity. We can use kinematic equations to describe the person’s position over time.
Let’s define the upward direction as positive and the downward direction as negative.
- Initial height (y₀): +2 meters (above the ground)
- Initial velocity (v₀): +50 m/s (upward)
- Acceleration due to gravity (a): -9.8 m/s² (downward)
- Target height (y): 0 meters (the ground)
The relevant kinematic equation relating position, initial position, initial velocity, acceleration, and time (t) is:
y = y₀ + v₀t + (1/2)at²
We want to find the time t when the person reaches the ground, so we set y = 0:
0 = 2 + 50t + (1/2)(-9.8)t²
This simplifies to a quadratic equation in terms of t:
0 = 2 + 50t – 4.9t²
Rearranging into the standard quadratic form (at² + bt + c = 0):
-4.9t² + 50t + 2 = 0
or
4.9t² – 50t – 2 = 0
We can solve for t using the quadratic formula: t = [-b ± sqrt(b² – 4ac)] / (2a)
Here, a = 4.9, b = -50, and c = -2.
t = [ -(-50) ± sqrt((-50)² – 4 * 4.9 * -2) ] / (2 * 4.9)
t = [ 50 ± sqrt(2500 + 39.2) ] / 9.8
t = [ 50 ± sqrt(2539.2) ] / 9.8
t = [ 50 ± 50.39 ] / 9.8
This gives two possible solutions for t:
t₁ = (50 + 50.39) / 9.8 = 100.39 / 9.8 ≈ 10.24 seconds
t₂ = (50 – 50.39) / 9.8 = -0.39 / 9.8 ≈ -0.04 seconds
Since time cannot be negative in this context (the motion starts at t=0), the physically meaningful answer is the positive one. Therefore, it takes approximately 10.24 seconds for the person to hit the ground after being thrown upward from a height of 2 meters.