Question

Use energy conservation to find the approximate final speed of a basketball dropped from a height of $\mathbf{2}$m (roughly the height of a professional basketball player). Why don't you need to know the mass of the basketball?

Short answer

The final speed of the basketball is $6.26\mathrm{m}/\mathrm{s}$. The final velocity does not dependent on mass, so the mass of basketball is not required.

Step-by-step solution

Relation between the ball’s potential and kinetic energy

If the ball is dropped from a height of $2\mathrm{m}$determines the approximate final speed of the ball.

Apply the energy conservation law. Explain why the value of mass is not needed.

Energy cannot be created or destroyed, according to the rule of conservation of energy. It can only be transferred from one type of energy to another.

The ball generates potential energy while losing kinetic energy as it hits the ground. As a result, the kinetic energy loss can be calculated using the formula,

role="math" localid="1657704999970" $\mathit{K}\mathit{E}\mathbf{=}{}_{\mathbf{2}}{}^{\mathbf{1}}\mathit{m}{\mathit{v}}^{\mathbf{2}}$

Further, the gain in potential energy can be expressed as,

$\mathit{P}\mathbf{}\mathit{E}\mathbf{=}\mathit{m}\mathbf{}\mathit{g}\mathbf{}\mathit{h}\mathbf{}$

As per the law of conservation,

$\mathit{K}\mathbf{}\mathit{E}\mathbf{=}\mathit{P}\mathbf{}\mathit{E}\mathbf{}\mathbf{}$

Determine the formula from the relation for speed

If we break down the formula above, we can discover the formula for calculating speed.

$\mathit{K}\mathit{E}\mathbf{=}\mathit{P}\mathit{E}\frac{\mathbf{1}}{\mathbf{2}}\mathit{m}{\mathit{v}}^{\mathbf{2}}$

$KE=mgh\frac{1}{2}{v}^2$

$=\mathrm{ghv}$

$=\sqrt{2gh}$

$g$is the acceleration due to gravity is equal to $9.8\mathrm{m}/{\mathrm{s}}^2$and $\mathrm{h}$is the height. Substitute the given values to the formula.

$v=\sqrt{2gh}$

$=\sqrt{2\left(9.8\mathrm{m}/{\mathrm{s}}^2\right)(2.0\mathrm{m})}$

$=6.26\mathrm{m}/\mathrm{s}$

Therefore, the approximate final speed of the ball is $6.26\mathrm{m}/\mathrm{s}$ Because the final speed is independent of the ball's mass, the mass is not required. That is, we can still calculate the speed value without it.