P2.1 - Motion

• Distance and displacement

  • They both measure how far something has travelled, but displacement also says which direction something has travelled in.
  • For example, you could say a car has travelled a distance of 10 m, but it has a displacement of 10 m north.
• Velocity

  Velocity is a more useful measure of motion, because it describes both the speed and direction. E.g. velocity = 30 mph due north.
• Scalar


  • Quantities like speed and distance, that are only numbers, are called scalar quantities.
  • Scalar quantities: speed, distance, mass, time, etc.
• Vector

  • Quantities like velocity and displacement, that have a direction as well, are vector quantities.
  • Vector quantities: velocity, displacement, force, acceleration, etc.
• Positive and negative direction in vectors

  • When we use vectors, we often talk about there being a positive and a negative direction.
  • E.g. a car moving in one direction could have a velocity of 3 m/s, but moving in the opposite direction it will have a velocity of -3 m/s. In this example, the car has a speed of 3 m/s in both directions.
  • You can often pick a positive direction that makes the calculations easier.
• Speed, Distance and Time Formula:

  distance travelled (m) = speed (m/s) x time (s)
• Converting units:

  To convert 16 km into m: multiply by 1000 so 16 × 1000 = 16 000 m

  To convert 22 ms into seconds: divide by 1000 so 22 / 1000 = 0.022 s
• Converting units:

  To convert 8 hr into s: Multiply 8 by 60 to find the number of minutes so 8 x 60 = 480 minutes

  Then multiply 480 minutes by 60 to find the number of seconds so 480 × 60 = 28 800 s
• SI Unit:

  The Sl unit of distance is the metre, and the Sl unit of time is the second, but you usually use miles and hours when you measure speed in a car.

  There are 1000 m/km-this means there are 1000 metres in one kilometre. There are 1609 m/mile. There are 3600 s/hour - this means there are 3600 seconds in one hour.
• Converting from mph to m/s - Example question:

  Convert 30 mph to m/s.

  • Step 1: Convert the miles to metres: 30 miles = 30 miles x 1609 m/mile = 48270 m
  • Step 2: Convert hours to seconds. 1 hour = 1 hour x 3600 s/hour = 3600 s
  • Step 3: Use the equation for speed to find the answer. speed = distance / time so 48 270 m / 3600 s = 13 m/s (2 sig figs)
• Acceleration:

  • Acceleration is the rate of change of velocity.
  • This change in velocity can be a change in speed or a change in direction or both.
  • The equation for calculating acceleration: acceleration (m/s²) = change in velocity (m/s) time (s)
  • Acceleration is like velocity - it's a vector and so can have a positive or negative value.
  • If an object has a negative acceleration, it is either slowing down (decelerating), or speeding up in the negative direction.
• Change in velocity:

  • To calculate the change in velocity, you must always do final velocity - initial velocity.
  • An object travelling in a circle at a constant speed has a changing velocity (because it's always changing direction), so it's always accelerating.
• Uniform acceleration:

  Uniform acceleration means that the acceleration is constant (does not change), you can use the following equation:

  (final velocity)² (m/s)2 - (initial velocity)² (m/s)2 = 2 × acceleration (m/s²) X distance (m)

  or v² - u² = 2 x a x d
• Investigating the motion of a trolley on a ramp:

  1. Mark a line on the ramp just before the first light gate - to make sure the trolley starts from the same point each time.
  2. Measure the distances between light gates 1 and 2, and 2 and 3.
  3. As the trolley rolls down the ramp it will accelerate. When it reaches the runway, it will travel at a constant speed (ignoring any friction).
  4. Each light gate will record the time when the trolley passes through it.
• Calculations for trolley experiment:

  • The time it takes to travel between gates 1 and 2 is used to find the average speed on the ramp, and between gates 2 and 3 gives the speed on the runway (speed = distance / time).
  • The acceleration of the trolley on the ramp is acceleration = change in speed / time, with the following values:
  • the initial speed of the trolley (= 0 m/s),
  • the final speed of the trolley, which equals the speed of the trolley on the runway (ignoring friction).
  • the time it takes the trolley to travel between light gates 1 and 2.
• Variations to do on trolley experiment:

  • The trolley's acceleration on the ramp and its final speed on the runway will increase when the angle of the ramp increases, or the amount of friction between the ramp and the trolley decreases.
  • Increasing the distance between the bottom of the ramp and where the trolley is released will also increase the final speed of the trolley.
• How to measure distance and Time:

  However using a stopwatch involves human error due to reaction times

  A) ruler

  B) stopwatch

  C) light gates

  D) ruler

  E) light gates

  
• Distance-Time (d-t) Graphs:

  • d-t Graphs tell you how far something has travelled. The different parts of a d-t graph describe the motion of an object:
  • The gradient (slope) at any point gives the speed of the object.
  • Flat sections are where it's stopped.
  • A steeper graph means it's going faster.
  • Curves represent acceleration.
  • A steepening curve means it's speeding up (increasing gradient).
  • A levelling off curve means it's slowing down (decreasing gradient).
• d-t graph:

  A) steady speed

  B) stopped

  C) accelerating

  D) decelerating

  
• The Speed of an Object Found From a Distance-Time Graph:

  • The gradient of a distance-time graph at any point is equal to the speed of the object at that time.
  • If the graph is a straight line, the gradient at any point along the line is equal to change in the vertical / change in the horizontal.
  • If the graph is curved, draw a tangent to the curve at that point, and then find the gradient of the tangent.
  • You can also calculate the average speed of an object when it has non-uniform motion (it's accelerating) by dividing the total distance travelled by the time it takes to travel that distance.
• Velocity-Time (v-t) graphs:

  • v-t graphs can be used to find acceleration
  • Gradient = acceleration.
  • Flat sections represent steady velocity.
  • The steeper the graph, the greater the acceleration or deceleration.
  • Uphill sections (/) are acceleration.
  • Downhill sections (\) are deceleration.
  • A curve means changing acceleration.
  • The area under any section of the graph (or all of it) is equal to the distance travelled in that time interval.
• v-t graph:

  A) constant acceleration

  B) steady velocity

  C) increasing acceleration

  D) steady velocity

  E) constant deceleration

  
• You can find the acceleration, velocity and distance travelled from a velocity-time graph:


  • The Acceleration = gradient = change in the vertical / change in the horizontal.
  • The velocity at any time is simply found by reading the value off the velocity axis.
  • The distance travelled in any time interval is equal to the area under the graph. For example, the distance travelled between t = 8 s and t = 10 s is equal to the shaded area, which is 10 m (5 m/s x 2 s)
• Use the Counting Squares Method to find the area under the graph


  • If an object has an increasing or decreasing acceleration, the graph is curved. You can estimate the distance travelled from the area under the graph by counting squares.
  • You need to find out how much distance one square of the graph paper represents.
  • To do this, multiply the width of square (in seconds) by the height of one square (in metres per second). Then you just multiply this by the number of squares under the graph.
  • If there are squares that are partly under the graph, you can add them together to make whole squares.
• Equation to calculate kinetic energy (energy in a kinetic store)

  kinetic energy (J) = 0.5 x mass (kg) x (speed (m/s))2