Final answer:
To achieve 160 J of potential energy, a 2.0 kg basketball must be thrown upward at an initial speed of approximately 12.67 m/s, calculated using the formula for gravitational potential energy and the kinematic equation for motion under constant acceleration.
Step-by-step explanation:
To find out how fast a 2.0 kg basketball must be thrown upward to achieve 160 J of potential energy, we need to first understand that the potential energy (PE) of an object elevated at a height h is given by the formula PE = mgh, where m is the mass of the object, g is the acceleration due to gravity (approximately 9.81 m/s² on Earth), and h is the height above the reference level.
We can rearrange the formula to solve for h: h = PE / (mg). Substituting the given values, we have h = 160 J / (2.0 kg × 9.81 m/s²) = 8.157 m.
Now, to determine the initial velocity (v) the basketball needs to reach this height, we can use the kinematic equation for an object's upward motion under constant acceleration (which, in this case, is gravity): v² = 2gh. Solving for v gives us v = √(2 × g × h). Substituting the values we have, we get v = √(2 × 9.81 m/s² × 8.157 m) = 12.67 m/s.
Therefore, the basketball must be thrown upward at an initial speed of approximately 12.67 m/s to achieve 160 J of potential energy.