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. (a) What is the speed of the putty just before it strikes the ceiling?

Verified step by step guidance

1. Identify the initial velocity (v_i) of the putty, which is 9.50 m/s upward, and the displacement (s) from the hand to the ceiling, which is 3.60 m upward.
2. Recognize that the acceleration due to gravity (a) is acting downward at approximately 9.81 m/s^2. Since the putty is moving upward against gravity, this acceleration will be negative in your calculations (-9.81 m/s^2).
3. Use the kinematic equation v_f^2 = v_i^2 + 2as to find the final velocity (v_f) of the putty just before it strikes the ceiling. Here, v_f is the final velocity, v_i is the initial velocity, a is the acceleration, and s is the displacement.
4. Substitute the known values into the equation: v_f^2 = (9.50 m/s)^2 + 2(-9.81 m/s^2)(3.60 m).
5. Solve the equation for v_f to find the speed of the putty just before it strikes the ceiling. Remember to take the square root of the result from the previous step to find v_f, and consider only the positive root since we are interested in the magnitude of the velocity.

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, and acceleration. In this scenario, kinematic equations can be used to relate the initial speed of the putty, the distance it travels, and its final speed just before hitting the ceiling.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the case of the putty, its initial kinetic energy as it leaves your hand is converted into gravitational potential energy as it rises. Just before it strikes the ceiling, this potential energy is converted back into kinetic energy, allowing us to calculate its speed.

Gravitational Acceleration

Gravitational acceleration is the acceleration of an object due to the force of gravity, typically denoted as 'g' and approximately equal to 9.81 m/s² near the Earth's surface. This constant affects the motion of the putty as it ascends and descends. It plays a crucial role in determining how the speed of the putty changes as it moves upward against gravity until it reaches its maximum height.