Calculus In PhysicsCalculus In Physics

By: May CheungBy: May Cheung

One-Dimensional MotionOne-Dimensional Motion

Examples:Examples: a car moving on a straight road a car moving on a straight road a person walking down a hallway a person walking down a hallway a sprinter running on a straight race course a sprinter running on a straight race course dropping a pencil dropping a pencil throwing a ball straight up throwing a ball straight up a glider moving on an air track a glider moving on an air track and many others... and many others...

DerivativesDerivatives

The position graph The position graph s(t), where x is s(t), where x is time and the y is distancetime and the y is distance

The velocity graph The velocity graph v(t), is the derivative v(t), is the derivative of the position graph, based on how of the position graph, based on how quickly the distance is changing quickly the distance is changing

The acceleration graph The acceleration graph a(t), is the a(t), is the derivative of the velocity graph, based on derivative of the velocity graph, based on how quickly the velocity is changinghow quickly the velocity is changing

PositionPosition

This graph records the distance that is This graph records the distance that is traveltravel

Let’s use the example of a person that is Let’s use the example of a person that is walkingwalking

the distance either increases or decreases the distance either increases or decreases relative to where the starting point is but to relative to where the starting point is but to make things easier right (forward) is make things easier right (forward) is positive and left (backward) is negativepositive and left (backward) is negative

VelocityVelocity this graph based on the slope of the position this graph based on the slope of the position

graph meaning how slow or how fast the person graph meaning how slow or how fast the person is travelingis traveling

A positive slope – the person is walking to the A positive slope – the person is walking to the right and the distance is increasingright and the distance is increasing

An increase in the positive slope - the person is An increase in the positive slope - the person is walking faster so the distance is increasing at a walking faster so the distance is increasing at a faster ratefaster rate

A decrease in the positive slope – the person is A decrease in the positive slope – the person is walking slower so the distance is increasing at a walking slower so the distance is increasing at a slower rateslower rate

Zero slope – the person has walking and no Zero slope – the person has walking and no distance has been traveleddistance has been traveled

Cont.Cont. Zero slope – the person has stopped and no Zero slope – the person has stopped and no

distance has been traveleddistance has been traveled A negative slope - the person is walking in the A negative slope - the person is walking in the

negative direction (left) so the distance is negative direction (left) so the distance is decreasing decreasing

An increase in the negative slope – the person is An increase in the negative slope – the person is walking faster in the left direction so the distance walking faster in the left direction so the distance is decreasing at a faster rateis decreasing at a faster rate

An decrease in the negative slope – the person An decrease in the negative slope – the person is walking slower in the left direction so the is walking slower in the left direction so the distance is decreasing at a slower ratedistance is decreasing at a slower rate

AccelerationAcceleration This graph is based on the slope of the velocity meaning This graph is based on the slope of the velocity meaning

how fast or how slow the person is changing his pacehow fast or how slow the person is changing his pace For example : walking to walking faster as opposed to For example : walking to walking faster as opposed to

walking to runningwalking to running a positive slope – the person changes from walking to a positive slope – the person changes from walking to

running, increasing the pace of its travel at a exponential running, increasing the pace of its travel at a exponential pattern (3 ft/sec to 9 ft/sec to 81 ft/sec)pattern (3 ft/sec to 9 ft/sec to 81 ft/sec)

Zero slope – the person changes from walking to walking Zero slope – the person changes from walking to walking faster at a consistent pattern (2 ft/sec to 4 ft/sec to 6 faster at a consistent pattern (2 ft/sec to 4 ft/sec to 6 ft/sec)ft/sec)

a negative slope – the person changes from running to a negative slope – the person changes from running to walking, decreasing the pace of its travel at an walking, decreasing the pace of its travel at an exponential pattern (81 ft/sec to 9 ft/sec to 3 ft/sec)exponential pattern (81 ft/sec to 9 ft/sec to 3 ft/sec)