You throw a glob of putty straight up toward the ceiling, which is $3.60$ m above the point where the putty leaves your hand. The initial speed of the putty as it leaves your hand is $9.50$ m/s. How much time from when it leaves your hand does it take the putty to reach the ceiling?

검증된 단계별 안내

Identify the known values: the initial velocity $ v_0 = 9.50 \text{ m/s} $, the displacement $ s = 3.60 \text{ m} $, and the acceleration due to gravity $ a = -9.81 \text{ m/s}^2 $ (negative because it acts downward).

Use the kinematic equation for displacement: $ s = v_0 t + \frac{1}{2} a t^2 $. Substitute the known values into this equation: $ 3.60 = 9.50 t + \frac{1}{2} (-9.81) t^2 $.

Rearrange the equation to form a quadratic equation: $ 0 = -4.905 t^2 + 9.50 t - 3.60 $.

Solve the quadratic equation using the quadratic formula: $ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = -4.905 $, $ b = 9.50 $, and $ c = -3.60 $.

Calculate the discriminant $ b^2 - 4ac $ to ensure it is non-negative, then solve for $ t $ using the quadratic formula to find the time it takes for the putty to reach the ceiling.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Kinematics Equations

Kinematics equations describe the motion of objects without considering the forces that cause the motion. In this problem, the equation v = u + at can be used to find the time taken for the putty to reach the ceiling, where v is the final velocity, u is the initial velocity, a is the acceleration due to gravity, and t is the time.

Acceleration due to Gravity

Acceleration due to gravity is a constant force acting on objects near the Earth's surface, typically approximated as 9.81 m/s² downward. This concept is crucial for calculating the motion of the putty as it travels upward against gravity, slowing down until it reaches the ceiling.

Projectile Motion

Projectile motion involves the movement of an object thrown into the air, subject to gravitational acceleration. Understanding this concept helps in analyzing the vertical motion of the putty, as it is projected upwards and decelerates until it reaches its peak height at the ceiling.