# Question 04812

Jun 16, 2015

You lack an essential piece of information to be able to solve this problem.

#### Explanation:

This is an interesting problem, but it can't be answered because you don't know how deep the object penetrates the surface upon impact.

The information you have allows you to determine the impact velocity of the object quite easily. However, in order to determine the impact force, you need to determine the acceleration the object has after the impact.

To make the calculations easier, convert pounds to kilograms and feet to meters.

1cancel("lb") * "1 kg"/(2.2046cancel("lbs")) = "0.454kg"

and

6cancel("feet") * "1 m"/(3.2808cancel("feet")) = "1.83 m"

You don't need to calculate the time of the fall, so you can determine the impact velocity by using

v_"impact"^2 = underbrace(v_0^2)_(color(blue)("=0")) + 2* g * h

${v}_{\text{impact"^2 = 2 * g * h => v_"impact}} = \sqrt{2 \cdot g \cdot h}$

${v}_{\text{impact" = sqrt(2 * 9.8 * 1.83) = "6.00 m/s}}$

Now, let's assume that the object penetrates the surface and stops after 10 cm, which is equivalent to 0.1 m. You can use the same general equation to determine the stopping acceleration, ${a}_{\text{stop}}$.

underbrace(v_"final"^2)_(color(red)("=0")) = underbrace(v_0^2)_(color(blue)(=v_text(impact)^2)) - 2 * a_"stop" * underbrace(x)_(color(green)("stopping distance"))#

${a}_{\text{stop" = v_"impact}}^{2} / \left(2 x\right)$

${a}_{\text{stop" = (2 * g * h)/(2x) = (cancel(2) * 9.8 * 1.83)/(cancel(2) * 0.1) = 179.3 "m"/"s}}^{2}$

This means that the impact force was

${F}_{\text{impact" = m * a_"stop" = "0.454 kg" * 179.3"m"/"s"^2 = "81.4 N}}$

The interesting thing to notice here is that the smaller the penetration distance, the bigger the impact force.

For example, let's assume that you drop this object on another surface and that the stopping distance is now 100 times smaller, i.e. 0.0001 m. Both the stopping acceleration, and the impact force would be 100 times greater.

${a}_{\text{stop" = (g * h)/x = (9.8 * 1.83)/(0.0001) = 179,340"m"/"s}}^{2}$

The impact force would be

${F}_{\text{impact" = m * a = "0.454 kg" * 179,340"m"/"s"^2 = "81,420 N}}$