LINEAR MOTION Advanced Higher Physics

Calculus Methods

Remember: Velocity and Acceleration Are VECTORS!!

Deriving the equations for uniform acceleration

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Horizontal Motion The kinematic relationships can be used to solve problems of motion in one dimension with constant acceleration. In all cases we will be ignoring the effects of air resistance.

Example A car accelerates from rest at a rate of 4.0 m s What is its velocity after 10 s? 2. How long does it take to travel 72 m? 3. How far has it travelled after 8.0 s?

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Tips The same strategy should be used in solving all of the problems for constant acceleration. list the data given to you in the question. It is often useful to make a sketch diagram with arrows, to ensure that any vector quantities are being measured in the correct direction.

Vertical Motion When dealing with freely-falling bodies on Earth, the acceleration of the body is the acceleration due to gravity, = 9.8 m s -2. If any force other than gravity is acting in the vertical plane, the body is no longer in free-fall, and the acceleration will take a different value. Problems should be solved using exactly the same method we used to solve horizontal motion problems.

Vertical Motion Example A student drops a stone from a second floor window, 15 m above the ground. 1. How long does it take for the stone to reach the ground? 2. With what velocity does it hit the ground?

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Tips Difficulties sometimes occur when the initial velocity is directed upwards, for example if an object is being thrown upwards from the ground. To solve such a problem, it is usual to take the vertical displacement as being positive in the upwards direction. The velocity vector is then positive when the object is travelling upwards, and negative when it is returning to the ground. In this situation the acceleration is always negative, as it is always directed towards the ground.

Graphs for motion with constant velocity The graphs below all represent motion with a constant velocity: You will also recall from Higher that the area under a velocity time graph is equal to the displacement. Similarly, it is worth noting that the area under an acceleration time graph is equal to the change in velocity.

Graphs for motion with constant positive acceleration The graphs below all represent motion with a constant positive acceleration: Note that since acceleration and velocity are the same sign, the object is speeding up.

Graphs for motion with constant negative acceleration The graphs below all represent motion with a constant negative acceleration: Note that since the acceleration and the velocity are the opposite sign, the object is slowing down.

Example 1 For the velocity - time graph shown, calculate: 1. The acceleration from 2.0 s to 8.0 s. 2. The displacement after 8.0 s.

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Example 2 Find the change in velocity in: 1. The first 5 seconds. 2. Between 0 and 30 seconds.

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Objects in Free fall

The acceleration due to gravity can also be found by considering the displacement-time graph. The gradient of the graph is increasing since the instantaneous velocity increases. The instantaneous velocity at a given time can be found by drawing a tangent to the graph and determining its gradient. To sketch a tangent to the graph, a straight line must be drawn such that it touches the curve only at one point.

Objects in free fall

Graphical methods for non-uniform acceleration So far we have assumed uniform (constant) acceleration. In other words, the change in velocity with time has been constant. For non-uniform acceleration, the change of velocity with time is not constant throughout the motion. So the slope of the velocity-time graph will be changing and it may have curved sections. Therefore, to find the acceleration at a given point in time, you need to draw a tangent to the curve at that point and then calculate its gradient.

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Examples of Varying Acceleration

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Summary You should now be able to use calculus notation to: represent velocity as the rate of change of displacement with respect to time; represent acceleration as the rate of change of velocity with respect to time; represent acceleration as the second differential of displacement with respect to time; derive the three kinematic relationships from the calculus definitions of acceleration and velocity apply these equations to describe the motion of a particle with uniform acceleration moving in a straight line; You should be able to state that/use: the gradient of a graph can be found by differentiation; the gradient of a displacement-time graph at a given time is the instantaneous velocity; the gradient of a velocity-time graph at a given time is the instantaneous acceleration; the area under a graph can be found by integration; for a velocity-time graph the displacement can be found by integrating between the limits; for an acceleration-time graph the change in velocity can be found by integrating between the limits; calculus methods to solve problems based on objects moving in a straight line with varying acceleration.

Tutorial Questions You can now complete all of tutorial 1