Distance, Speed and Acceleration

Acceleration A relationship between speed and time. It is the change in speed over the change in time. a =Δv/Δt Acceleration is measured in meters per second, per second OR m/s2 Basically, it is a measure of how an object is speeding up or slowing down.

If the relationship between change in speed and change in time remains the same throughout the acceleration, it is called constant acceleration. When acceleration varies over a period of time, we generally describe the object’s average acceleration. vav

p 384 Suppose you speed up on a motorcycle from rest (0 m/s) to 9.0 m/s in a time of 2.0 s. Your change in speed is 9.0 m/s and your average acceleration is calculated as follows: Δv = 9.0 m/s Δt = 2.0 s a=? aav=Δv/Δt =9.0m/s = 4.5 m/s = 4.5 m/s2 2.0 s s “ 4.5 meters per second squared”

p 385 Myriam Bedard accelerates at an average 2.5 m/s2 for 1.5 s. What is her change in speed at the end of 1.5 s? aav = 2.5 m/s2 Δt = 1.5 s Δv = ? aav = Δv/Δt Δv = aavΔt = 2.5 m/s2 x 1.5 s = 3.8 m/s Bedard’s change in speed is 3.8 m/s.

Refining the equation In a car, we usually know our initial speed vi From this initial speed, we accelerate to a final speed. vf Both the initial speed and final speed affect the change in speed. We can rewrite the equation to be more specific: aav = Δv/Δt becomes

p 386 A skiier is moving at 1.8 m/s at the top of a hill. 4.2 s later she is travelling at 8.3 m/s. What is her average acceleration? vi = 1.8 m/s Δt = 4.2 s v2 = 8.3 m/s aav = ? a = (8.3 – 1.8) m/s 4.2 s = 1.5 m/s2

Rearranging This new equation can be rearranged to find: Initial speed Final speed The change in time

Graphing Acceleration A speed-time graph shows acceleration. t (s) v (m/s) 2 5 4 10 6 15 8 20 25 12 30 Speed (m/s) Time (s)

Slope and Acceleration A positive slope represents positive acceleration (speeding up). A negative slope represents negative acceleration (slowing down). No slope represents constant speed.

Area Under the Line Multiplying the two variables describes a geometric shape in the graph. From this shape, we can get an idea of the distance travelled. We can find the distance by simply finding the area under the line. Speed (m/s) Time (s)

p 392, Sample problem 1 A boat travels at full throttle for 1.5h. Using the graph, find the distance travelled.

p 392, Sample Problem 2 Galieleo rolls a ball down a long grooved inclined plane. According to the speed-time graph, what is the distance travelled in 6.0 s?

Distance-Time vs Speed-Time

Comparing Graphs Distance - time Speed - time A straight positive slope A straight horizontal line

Instantaneous Speed Instantaneous speed is the speed at any particular point in time From a speed-time graph, we can read the instantaneous speed at any point along the line.

Constant vs Instantaneous On a Distance-time graph, if the speed is constant then the instantaneous speed is the same at any time, and is equal to the speed and at that time. More work is required when speed is not constant.

When speed is not constant: The first step is to draw a tangent across the point in time that you wish to find the speed for. You then calculate the slope for the new tangent you have drawn, giving you the instantaneous speed at that point.

Find the instantaneous speed at 2 seconds.

Find the instantaneous speed at 8 seconds.

Find the instantaneous speed at 1 second.