Interpreting Change in velocity from the Curve

Since the curve is inclined in the upward direction, horizontally and in the downward direction, the change in velocity of the body should be assessed from the curve/sketch in the three situations.

Taking a point on the curve slanting upwards (Point A marked in the sketch),

• Positive Velocity
  
  The tangent at any point on the curve while it is slanting upwards would have an inclination less than 90^o. Therefore, the velocity would be positive i.e. directed in the positive direction.
  

  For any point on the curve oriented in the upward direction, velocity is directed in the positive direction.

  
  


• Increase in Velocity
  
  Where the curve slants to the right (i.e. towards the horizontal or away from the vertical) the inclination of the tangent along the points on the curve would decrease. Therefore, the velocity which is the slope of the tangent would decrease.
  

  Where the curve tilts towards the right the velocity decreases.

  
  


• Decrease in Velocity
  
  Where the curve slants to the left (i.e. towards the vertical or away from the horizontal) the inclination of the tangent along the points on the curve would increase. Therefore, the velocity which is the slope of the tangent would increase.
  

  Where the curve tilts towards the left the velocity increases.

  
  


• No Change in Velocity
  
  Where the curve moves along the same direction as the tangent at that point, the velocity would not change.
  

  Where the curve is a straight path, the velocity is constant.

  
  


• Zero Velocity
  
  Where the curve is inclined along the horizontal, the velocity would be zero.

Taking a point on the curve slanting downwards (Point P marked in the sketch),

• Negative Velocity
  
  The tangent at any point on the curve while it is slanting downwards would have an inclination greater than 90^o. This implies that the slope of the tangent at any point on the curve which is oriented in the downward direction is negative. Therefore, the velocity would be negative i.e. directed in the negative direction.
  

  For any point on the curve oriented in the downward direction, velocity is directed in the negative direction.

  
  


• Increase in Velocity
  
  Where the curve slants to the right (i.e. towards the vertical or away from the horizontal) the inclination of the tangent along the points on the curve would decrease. Therefore, the velocity which is the slope of the tangent would increase. [For inclination above 90^o, the slope (tan θ) increases with a decrease in inclination]
  

  Where the curve tilts towards the right the velocity increases.

  
  


• Decrease in Velocity
  
  Where the curve slants to the left (i.e. towards the horizontal or away from the vertical) the inclination of the tangent along the points on the curve would increase. Therefore, the velocity which is the slope of the tangent would decrease. [For inclination above 90^o, the slope (tan θ) decreases with an increases in inclination]
  

  Where the curve tilts towards the left the velocity increases.

  
  


• No Change in Velocity
  
  Where the curve moves along the same direction as the tangent at that point, the velocity would not change.
  

  Where the curve is a straight path, the velocity is constant.

  
  


• Zero Velocity
  
  Where the curve is inclined along the horizontal, the velocity would be zero.

Velocity Curve :: Slopes of Tangent and Secants may be positive or negative

Where the velocity is directed in the positive direction i.e. upwards the inclination of the tangent to the curve is acute giving a positive slope.

Similarly, where the velocity is directed in the negative direction i.e. downwards the inclination of the tangent to the curve is obtuse giving a negative slope.

The tangent and the secant lines would therefore be lines with an inclination between zero and 180^o.

Uniform Velocity

Uniform velocity is indicated by a straight line curve/sketch. Between two points, where the curve takes the same direction as the tangent at the initial point we can say that the velocity is uniform between those two points.