Kinematics
Kinematics deals with situations involving motion. S=distance/displacement. If s=0, object is at starting point +- relates to position from a certain point (above/right vs below/left). V=velocity=rate at which distance changes over time +- relates to direction of motion (forwards/up vs down/back) If v=0, object is stationary A=acceleration=rate at which velocity changes over time +- relates to speeding up/slowing down. If a=0, object is moving at constant speed t=time Initial values: when t=0
Differentiate Integrate (add one to power/divide by new power) S displacement V velocity Acceleration A Differentiate Integrate (add one to power/divide by new power)
For example… The van’s velocity at the end of the 6th second is 10metres/second.
More examples (try on your own then check below for worked answers) A stone is thrown vertically upwards into the air. Its height above ground (in metres) is given by: s(t)=30t-5t2 Calculate the initial velocity Calculate the maximum height reached by the object When does the object return to the ground? The velocity (m/s)of an object t seconds after it started from the origin is given by: v(t)=3t2-14t-5 The object is travelling in a straight line in an East-West direction. Give formulae for acceleration & distance after t seconds. Calculate the initial velocity. In which direction is the object moving? When is the object at rest? Calculate the minimum velocity. Interpret this answer. How far did the object travel in the 8th second?
1. A stone is thrown vertically upwards into the air 1. A stone is thrown vertically upwards into the air. Its height above ground (in metres) is given by: s(t)=30t-5t2 Calculate the initial velocity Calculate the maximum height reached by the object When does the object return to the ground? v=s’=30-10t when t=0, v=30m/s Max height occurs when v=0 v=30-10t=0 t=3sec s(3)= 30t-5t2 =45m : max height is 45m. Object returns to ground when s=0 30t-5t2 =0 5t(6-t)=0 t=0 or 6s The object returns to the ground after 6 seconds (t=0 is the initial time on the ground)
V(0)= 3*02-14*0-5 =-5m/s The object is moving west->east The velocity (m/s)of an object t seconds after it started from the origin is given by: v(t)=3t2-14t-5 The object is travelling in a straight line in an East-West direction. Give formulae for acceleration & distance after t seconds. Calculate the initial velocity. In which direction is the object moving? When is the object at rest? Calculate the minimum velocity. Interpret this answer. How far did the object travel in the 8th second? a=v’=6t-14 s= ∫ 3t2-14t-5.dt=3t3/3-14t2/2-5t+c. s=t3-7t2-5t (since s=0 when t=0, c=0) V(0)= 3*02-14*0-5 =-5m/s The object is moving west->east The object is at rest when v=0 3t2-14t-5=0 (3t+1)(t-5)=0 t=-1/3 or 5 : object is at rest after 5sec (-1/3 is before motion started) Minimum velocity occurs when a=0 a= 6t-14=0 t=14/6 sec (2.33sec) V= 3t2-14t-5 v(14/6)=-21.33m/s The minimum velocity is -21.33m/s During the 8th second, the object travels the difference of distance of the object from the origin at t=7 and t=8. At t=7, s(7)=73-7*72-5*7=-35m At t=8, s(8)= 83-7*82-5*8=24m the object moves from -35m to 24m during the 8th second, so travels a total of 59m. (alternatively, use definite integration between t=7&8)
More examples (try on your own then check below for worked answers) A stone is thrown vertically upwards into the air. Its height above ground (in metres) is given by: s(t)=30t-5t2 Calculate the initial velocity Calculate the maximum height reached by the object When does the object return to the ground? The velocity (m/s)of an object t seconds after it started from the origin is given by: v(t)=3t2-14t-5 The object is travelling in a straight line in an East-West direction. Give formulae for acceleration & distance after t seconds. Calculate the initial velocity. In which direction is the object moving? When is the object at rest? Calculate the minimum velocity. Interpret this answer. How far did the object travel in the 8th second?
More examples (try on your own then check below for worked answers) A stone is thrown vertically upwards into the air. Its height above ground (in metres) is given by: s(t)=30t-5t2 Calculate the initial velocity Calculate the maximum height reached by the object When does the object return to the ground? The velocity (m/s)of an object t seconds after it started from the origin is given by: v(t)=3t2-14t-5 The object is travelling in a straight line in an East-West direction. Give formulae for acceleration & distance after t seconds. Calculate the initial velocity. In which direction is the object moving? When is the object at rest? Calculate the minimum velocity. Interpret this answer. How far did the object travel in the 8th second?
More examples (try on your own then check below for worked answers) A stone is thrown vertically upwards into the air. Its height above ground (in metres) is given by: s(t)=30t-5t2 Calculate the initial velocity Calculate the maximum height reached by the object When does the object return to the ground? The velocity (m/s)of an object t seconds after it started from the origin is given by: v(t)=3t2-14t-5 The object is travelling in a straight line in an East-West direction. Give formulae for acceleration & distance after t seconds. Calculate the initial velocity. In which direction is the object moving? When is the object at rest? Calculate the minimum velocity. Interpret this answer. How far did the object travel in the 8th second?
More examples (try on your own then check below for worked answers) A stone is thrown vertically upwards into the air. Its height above ground (in metres) is given by: s(t)=30t-5t2 Calculate the initial velocity Calculate the maximum height reached by the object When does the object return to the ground? The velocity (m/s)of an object t seconds after it started from the origin is given by: v(t)=3t2-14t-5 The object is travelling in a straight line in an East-West direction. Give formulae for acceleration & distance after t seconds. Calculate the initial velocity. In which direction is the object moving? When is the object at rest? Calculate the minimum velocity. Interpret this answer. How far did the object travel in the 8th second?
More examples (try on your own then check below for worked answers) A stone is thrown vertically upwards into the air. Its height above ground (in metres) is given by: s(t)=30t-5t2 Calculate the initial velocity Calculate the maximum height reached by the object When does the object return to the ground? The velocity (m/s)of an object t seconds after it started from the origin is given by: v(t)=3t2-14t-5 The object is travelling in a straight line in an East-West direction. Give formulae for acceleration & distance after t seconds. Calculate the initial velocity. In which direction is the object moving? When is the object at rest? Calculate the minimum velocity. Interpret this answer. How far did the object travel in the 8th second?
More examples (try on your own then check below for worked answers) A stone is thrown vertically upwards into the air. Its height above ground (in metres) is given by: s(t)=30t-5t2 Calculate the initial velocity Calculate the maximum height reached by the object When does the object return to the ground? The velocity (m/s)of an object t seconds after it started from the origin is given by: v(t)=3t2-14t-5 The object is travelling in a straight line in an East-West direction. Give formulae for acceleration & distance after t seconds. Calculate the initial velocity. In which direction is the object moving? When is the object at rest? Calculate the minimum velocity. Interpret this answer. How far did the object travel in the 8th second?