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. (b) How much time from when it leaves your hand does it take the putty to reach the ceiling?

Verified step by step guidance

1. Identify the initial velocity (v_0) of the putty, which is 9.50 m/s, and the displacement (s) from the hand to the ceiling, which is 3.60 m. The acceleration due to gravity (a) is -9.8 m/s^2, acting downwards.
2. Use the kinematic equation for motion: s = v_0 t + \frac{1}{2} a t^2, where s is the displacement, v_0 is the initial velocity, a is the acceleration, and t is the time.
3. Substitute the known values into the equation: 3.60 = 9.50t + \frac{1}{2}(-9.8)t^2.
4. Rearrange the equation to form a standard quadratic equation: 4.9t^2 - 9.50t + 3.60 = 0.
5. Solve the quadratic equation for t using the quadratic formula: t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a = 4.9, b = -9.50, and c = 3.60. Choose the positive root for time.

Key Concepts

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, acceleration, and time. In this scenario, kinematic equations can be used to relate the initial velocity of the putty, the distance to the ceiling, and the time taken to reach that height.

Acceleration due to Gravity

Acceleration due to gravity is the rate at which an object accelerates towards the Earth when in free fall, typically denoted as 'g' and approximately equal to 9.81 m/s². When the putty is thrown upwards, it experiences a downward acceleration due to gravity, which affects its upward motion and ultimately determines how long it takes to reach the ceiling.

Projectile Motion

Projectile motion refers to the motion of an object that is thrown or projected into the air, subject to the influence of gravity. In this case, the putty's upward trajectory can be analyzed as a one-dimensional projectile motion problem, where the initial velocity, the height of the ceiling, and the effects of gravity are key factors in determining the time to reach the ceiling.