Graphs and equations of motion - 3.4

If we consider the area below the v vs t graph at any velocity, v at a time t, then the area is a triangle with value ____.

1⁄2vΔt

For an s vs t graph

= gradient of the x vs t graph

In this chapter we will specifically look at the relationships between displacement, time, velocity and acceleration. The relationship between force, mass and the quantities mentioned above will be explored when we investigate Newton's Laws.

The relationships between displacement, time, velocity and acceleration can be described by equations of motion. We will also explore these relationships graphically.

(excludes Δx)

vf = vi + aΔt

Solving problems When solving problems using equations of motion it is advisable to start by listing all the quantities that are given. It is often then only a case of identifying which equation can be used to solve the unknown. 1. A car travelling at 54 km/h brakes with an acceleration of 2 m/s2 opposite to its direction of motion. How far does the car travel from the moment it starts applying the brakes until it stops?

Solution: List all the quantities that are given: vi = 15 m/s vf = 0 a = -2m/s2 Δx = ? Identify the appropriate equation of motion, substitute the given values and solve:

2. Calculate how long it would take an object to fall from rest through a distance of 2 m in earth's gravitational field.

Solution: Take downward to be positive. vi = 0 Δx = 2 m g = 9,8 m/s2 Δt = ?

The acceleration-time graph would therefore be as follows:

The acceleration is zero for all times of the motion.

The displacement after time interval Δt can be calculated using an equation of motion as follows: (derivation)

The area below the v vs t graph therefore still indicates the change in displacement.

v1t1 in the graph represents the shaded area below the graph. According to the equation Δx = vΔt, Δt represents the time interval 0 to t1 and Δx represents the displacement, x1 at time t1. Therefore

X1=v1t1. That means X1 is equal to the area below the v vs t graph at time t1.

The shapes of graphs for uniform acceleration The following graphs represent a constant positive acceleration. The direction of motion is taken to be positive. A constant acceleration would mean a ____. The acceleration- time graph would therefore be as follows:

changing velocity.

Δx (s) - vi (u) - vf (v) - Δt - a -

displacement (change in position) initial velocity final velocity change in time acceleration

The equations reduce to ____ These are of course the equations we had for uniform velocity. The result is consistent with our expectation that _____

either vf = vi or Δx= viΔt. zero acceleration implies a uniform velocity.

Similarly Δv = aΔt. This means that the velocity is... The same point that is made in the box on the right could be made for the relationship between the area below an a vs t graph and the change in velocity.

equal to the area below the a vs t graph.

When solving problems where an object is accelerating because of earth's gravitational field, we use ____. The value for g is taken to be 9,8 m/s2 unless otherwise stated in the problem. Remember that the gravitational acceleration is ___

g as the symbol for acceleration instead of a downward towards the centre of earth.

If we change the subject of the formula for each equation it reveals another relationship between the graphs. Let's look at Δs = vΔt. The right hand side of the equation represents ____. This can easily be seen by looking at a v vs t graph for constant velocity.

the area below the velocity-time graph.

2. The area below the velocity-time graph over a certain time interval is equal to...

the change in displacement over that time interval.

4. The area below the acceleration-time graph over a certain time interval is equal to...

the change in velocity over that time interval.

The shapes of graphs for uniform velocity The following graphs represent a constant positive velocity, this means ____ A constant velocity would mean a ___ acceleration.

the direction of motion is taken to be positive. zero

The value of the velocity at a certain time is therefore equal to

the gradient of the displacement time graph at that time. The same would be true for the relationship between speed-time and distance-time graphs.

To draw the displacement-time graph we need to remember that the value of the velocity is equal to ____. The displacement-time graph therefore needs to have a ____

the gradient of the displacement-time graph constant positive gradient.

To draw to velocity-time graph we need to remember that the value of the acceleration is equal to _____. The velocity-time graph therefore needs to have a ____ gradient. As we stated that the direction of motion is positive, the graph needs to indicate _____

the gradient of the velocity- time graph constant positive positive values for velocity.

To draw to velocity-time graph we need to remember that the value of the acceleration is equal to ____ The velocity-time graph therefore needs to have a ____. As we stated that the direction of motion is positive, the graph needs to indicate ____

the gradient of the velocity- time graph. zero gradient positive values for velocity.

Similarly ___ indicated that the value of acceleration at a certain time is equal to

the gradient of the velocity-time graph at that time.

3. The gradient of a velocity-time graph at a certain time is equal to

the value of the acceleration at that time.

To summarise: 1. The gradient of a displacement-time graph at a certain time is equal to...

the value of the velocity at that time.

We also notice that the area below the acceleration-time graph is zero, meaning that ____. The area below the velocity-time graph for any time interval will be equal to____.

there can be no change in the velocity the change in displacement over that time interval

The equation s = vΔt describes the relationship between the quantities needed to describe ____. This equation does not show any relationship with acceleration and therefore we need other equations to describe the relationship between the quantities for ______. It is beyond the scope of your syllabus to know how these equations are derived.

uniform velocity. uniform acceleration.

Because the object is now accelerating, there are different ____. The different equations incorporate ____.

velocities at different times initial velocity, final velocity, average velocity or combinations of them.

To draw the displacement-time graph we need to remember that the value of the ____ is equal to the gradient of the displacement-time graph. The displacement-time graph therefore needs to have an ____ gradient, as the velocity values are increasing positive values.

velocity increasing positive

(excludes Δt)

vf² = vi² + 2aΔx

(excludes vf)

Δx = viΔt+ 1⁄2aΔt²