Newton's+Laws

__**Newtons Laws:**__ While this video complies with all three of Newton's laws, it can be analyzed most in depth with Newton's 2nd law.

Newtons 2nd law states that the sum of the Forces on an object are directly proportional to that objects acceleration and inversely proportional to the mass of the object. This is modeled by the equation ∑F = ma. This law is also given in terms of momentum; it states that the sum of the Forces on an object are directly proportional to that object's change in momentum and inversely proportional to the change in time. This is modeled by the equation ∑F = ∆p / ∆t.

∑F = ma ∑F = 100(kg) * 2.536(m/s^2) ∑F = 253.6N
 * ∑F = ma:** The easiest way to find the amount of force that Wes and Cooper apply on one another is to use the equation ∑F = ma. Unfortunately, while this equation is the simpler of the two, it can only really be used to find the force Wes applies on Cooper, the reason for this will be addressed later. To calculate force, Wesley's acceleration prior to the collision and Wesley's mass must be known. Wesley's mass was 100 kg at the time and using the slope from Wesley's Velocity / Time graph his acceleration prior to the collision is 2.536 m/s^2. Now it's just plug and chug.


 * ∑F = ∆p / ∆t:** As was stated above, Newtons second law can be modeled by two equations ∑F = ma and ∑F = ∆p / ∆t. Out of the two equations ∑F = ∆p / ∆t is a little more complicated, however it a much better equation to use. The reason for this is because both the force Wes applies on Cooper and the force Cooper applies on Wes can be found. This cannot be done with ∑F = ma because Cooper has no initial acceleration. If ∑F = ∆p / ∆t is used both forces can be found because it uses ∆p (momentum) and this can be calculated for Wes and Cooper. The expanded version of ∑F = ∆p / ∆t is ∑F = (mv-mv) / t, so in order to calculate force using this equation Cooper and Wesley's masses must be known, as well as each of their final and initial velocities, in addition the time between the initial and final velocities must be known. The methods for finding these values and the values themselves are as follows:

Time- Rather than using the graphs to find the time the video itself was used. It was found that the camera taking the video was operating at 15 frames per second. The collision itself took place over a total of 8 frames (and 8 data points) so the total time that elapsed during the collision was 8 (frames) / 15 (frames/s). The total time elapsed was .53(s).

Wesley's Initial Velocity- This value was found by looking at Wesley's velocity/time graph right before the collision. This value was 4.381(m/s).

Wesley's Final Velocity- This value was found by looking at Wesley's velocity/time graph using a point 8 frames(.53s) after the point for initial velocity. This value was found to be 2.951(m/s).

Wesley's Mass- Wesley's weight in pounds was found prior to the video and the 220(lbs) that he weighed was changed to 100(kg).

Cooper's Initial Velocity- This value was 0(m/s) because Cooper wasn't moving prior to the collision.

Cooper's Final Velocity- Because Cooper's initial velocity was 0 this value was found using Cooper's distance/time graph. The slope of the line between the point right before the collision and a point 8 frames(.53s) after was found. This value was used for Cooper's final velocity and is 2.093(m/s).

Cooper's Mass- Like Wesley's mass this was measured before hand in pounds and was converted to kilograms. 138(lbs) to 62.7(kg).

∑F = (mv-mv) / t ∑F = (100*2.951 - 100*4.381) / .53 ∑F = -143 / .53 ∑F = -269.8N Note: This value is negative because of direction.
 * Wesley's Applied Force:**

∑F = (mv-mv) / .53 ∑F = (62.7*2.093 - 0) / .53 ∑F = 131.23 / .53 ∑F = 247.6N
 * Cooper's Applied Force**


 * Percent Error:** To sum everything up for Newton 2nd law, percent error was calculated using all of the above values for force. We can do this because according to Newton's 3rd law all three values should be the same. The value that was calculated using the equation ∑F = ma as the "correct" value. This was done because there were only two variables, as opposed to five. Also, acceleration was found using the slope of a line, not just using a single point. There was a 6.39% error between Wesley's ∑F = ma and ∑F = ∆p / ∆t. There was only a 2.40% error between Wesley's ∑F = ma and Cooper's ∑F = ∆p / ∆t.