Motion

Key points

Speed and velocity

• Speed is the distance an object travels per unit time. “Per unit time” means “in each second”.
• v = s / t, where v is speed, s is distance travelled and t is time taken.
• The unit of speed is m/s (metres per second).
• To change km/h into m/s, multiply by 1000 and divide by 3600.
• Velocity is a speed in a stated direction. Two cars moving at 15 m/s in opposite directions have the same speed but different velocities.
• The speed of a car or a ferry usually changes during a journey. The average speed is the total distance travelled divided by the total time taken.
• The total time includes any time spent stopped.

Distance–time graphs

• The gradient (slope) of a distance–time graph is the speed.
• Gradient = change in distance ÷ change in time. Use two points on the line that are far apart.
• Calculate speed from the gradient only where the line is straight.
• Reading the shape:
  • horizontal line: the object is at rest (the distance does not change)
  • straight sloping line: constant speed. A steeper line means a higher speed.
  • curve getting steeper: the speed is increasing, so the object is accelerating
  • curve getting less steep: the speed is decreasing, so the object is decelerating
• You can use a table of data in the same way. Look at how the distance changes in equal time intervals:
  • no change in distance: at rest
  • the same increase in each interval: constant speed
  • a bigger increase in each interval than in the one before: accelerating
  • a smaller increase in each interval than in the one before: decelerating

Speed–time graphs

• Reading the shape:
  • horizontal line on the time axis (speed = 0): the object is at rest
  • horizontal line above the time axis: constant speed
  • line sloping upwards: the speed is increasing, so the object is accelerating
  • line sloping downwards: the speed is decreasing, so the object is decelerating
• A horizontal line on a speed–time graph does not mean the object is at rest. It means the speed is not changing.
• The area under a speed–time graph is the distance travelled.
• For constant speed, the area is a rectangle: distance = speed × time.
• For constant acceleration from rest, the area is a triangle. If the object does not start from rest, split the area into a rectangle and a triangle on top of it. Area of a triangle = ½ × base × height.
• The units check: (m/s) × s = m.
• The gradient of a speed–time graph is the acceleration. Extended
• A straight sloping line means constant acceleration. Extended
• A curved line means changing acceleration, because the gradient is changing. Extended
• For a curve that slopes upwards (speeding up): if the curve gets less steep, the acceleration is decreasing; if it gets steeper, the acceleration is increasing. Extended
• You can also spot constant or changing acceleration in a table of data. If the speed changes by the same amount in each equal time interval, the acceleration is constant. Extended

Acceleration and deceleration Extended

• Acceleration is the rate of change of velocity: the change in velocity divided by the time taken for that change. Extended
• a = Δv / Δt. The symbol Δ means “change in”. Extended
• The unit of acceleration is m/s². An acceleration of 2 m/s² means the velocity increases by 2 m/s every second. Extended
• Δv = final velocity − starting velocity. Extended
• A deceleration is a negative acceleration: the object is slowing down. In a calculation, Δv is negative, so a is negative. Extended
• For example, a deceleration of 3 m/s² is the same as an acceleration of −3 m/s². Extended

Falling objects

• The acceleration of free fall, g, is the acceleration of an object falling under gravity alone.
• Near the surface of the Earth, g is approximately constant. Its value is approximately 9.8 m/s².
• A uniform gravitational field has the same strength and the same direction at every point. In a small region near the Earth’s surface, the gravitational field is very nearly uniform. Extended
• Air resistance is a force from the air that opposes the motion of an object. It increases as the speed increases. Extended
• Falling without air resistance (for example, in a vacuum): Extended
  • The only force on the object is its weight.
  • Near the Earth’s surface, the object falls with constant acceleration g. Its speed increases by 9.8 m/s every second.
  • The speed–time graph is a straight line with a gradient of 9.8 m/s².
  • Every object has this same acceleration, whatever its mass.
• Falling in air, with air resistance (the upward push of the air, called upthrust, is so small that we ignore it): Extended
  1. The object is dropped from rest. At the start, its speed is zero, so there is no air resistance. The object accelerates at g.
  2. As the speed increases, the air resistance increases.
  3. The resultant force (weight − air resistance) gets smaller, so the acceleration gets smaller (see 1.5).
  4. In the end, the air resistance becomes equal to the weight. The resultant force is zero, so the acceleration is zero.
  5. The object now falls at a constant speed. This speed is called the terminal velocity.
• The speed–time graph for a fall with air resistance starts steep, curves and gets less steep, then becomes horizontal at the terminal velocity. Extended
• Sinking through a liquid (for example, a ball bearing in oil): the liquid gives a resistance force (drag) that increases with speed. The liquid also pushes up on the object with a force called upthrust, even when the object is not moving. Extended
• In a liquid, the resultant force is weight − upthrust − drag. As the speed increases, the drag increases, so the resultant force and the acceleration decrease. Extended
• The object reaches its terminal velocity when drag + upthrust = weight. Extended
• A liquid such as oil gives much more drag than air at the same speed. So an object falling through a liquid has a lower terminal velocity than in air. Extended
• When a skydiver opens a parachute, the air resistance suddenly becomes larger than the weight. The skydiver decelerates. Air resistance falls as the speed falls, until it equals the weight again. The skydiver then falls at a new, lower terminal velocity. Extended

Common mistakes

• Students write that a horizontal line on a speed–time graph means the object has stopped. / The mark scheme wants: constant speed. The object is at rest only when the line is on the time axis (speed = 0).
• Students find the average speed by adding the speeds and dividing by the number of speeds. / The mark scheme wants: total distance ÷ total time, including any time spent stopped.
• Students forget the ½ when they find the area of a triangle under a speed–time graph. / The mark scheme wants: area of a triangle = ½ × base × height.
• Students find a gradient by dividing one reading by another, from a line that does not start at the origin. / The mark scheme wants: change in distance ÷ change in time, from two points on the line.
• Students write a deceleration as a positive acceleration in a calculation. / The mark scheme wants: Δv = final − starting velocity, so a slowing object has a negative acceleration. Extended
• Students write that at terminal velocity there are no forces, or that air resistance is bigger than the weight. / The mark scheme wants: air resistance equals the weight, so the resultant force is zero and the acceleration is zero. Extended

Exam tips

• Sketch a graph: label both axes with the quantity and the unit. Show the shape clearly, for example a straight line or a curve that levels off.
• Plot a graph: put time on the x-axis, choose a scale that uses more than half the grid, plot small crosses, and draw one smooth line or curve of best fit.
• Describe the motion from a graph with exact phrases: “at rest”, “constant speed”, “accelerating”, “decelerating”. Give values from the graph if you can.
• Calculate a gradient: draw a large triangle on the line, at least half the length of the line. Write the values you read.
• Calculate a distance from a speed–time graph: split the area into rectangles and triangles. Add the parts.
• State the value of g: “approximately 9.8 m/s², and approximately constant near the Earth’s surface.”
• Determine the type of acceleration: “straight line = constant acceleration; curve = changing acceleration.” Extended
• Always give the unit of acceleration as m/s², and keep the minus sign for a deceleration. Extended
• A typical 1-mark answer: “Velocity is speed in a particular direction.”
• A typical 2-mark answer to “How do you find the distance travelled from a speed–time graph?”: “Find the area under the graph (1). Use the area of each rectangle and triangle and add them (1).”
• A typical 3-mark answer to “Explain why a falling skydiver reaches a terminal velocity”: “As the speed increases, the air resistance increases (1). The resultant force and the acceleration decrease (1). When the air resistance equals the weight, the resultant force is zero and the speed is constant (1).” Extended