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# 2.1 linear motion

## by John Kennedy, Head of Science at St Andrew's Cathedral School on May 18, 2011

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An overview to motion in a straight line (kinematics)

An overview to motion in a straight line (kinematics)

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## 2.1 linear motionPresentation Transcript

• Topic 2 - Mechanics 2.1 - Kinematics
• Velocity, Speed, Acceleration
• Kinematics is the study of linear motion.
• There are a number of key terms to define;
• Displacement – This is the distance from a given starting point to the current position in a straight line.
• Velocity – This is the speed of an object in a given direction
• Velocity is the rate of change of position
• Acceleration – This is the rate of change of velocity in a given direction.
• All three of these quantities are vectors.
• Average Speed and Velocity
• The average speed of an object is defined as:
• NB. It is the total distance and the total time that are important.
• The average velocity of an object is defined as:
• Average Speed and Velocity
• Calculate the average speed and velocity of:
• A car that travels at 30ms -1 East for 2 minutes before turning North and travelling at 40ms -1 for 5 minutes.
• A ball that travels at 10ms -1 for 15s before striking a wall and rebounding at 45 o to the original path at 8ms -1 for 5s.
• A 400m runner on a standard athletics track that completes the race in 52s.
• Instantaneous Motion
• Average values of displacement, velocity and acceleration give reasonable approximations of the behaviour of a system.
• They tell you what the system was like before and what it will be like afterwards.
• However, they do lack detail.
• They do not tell you how and object got from A to B only that it did.
• Instantaneous values for displacement, velocity and acceleration give a much clearer picture.
• Instantaneous Motion
• Instantaneous displacement (position) is often simply measured.
• It can be recorded by various means including using cameras.
• Instantaneous velocity is calculated from a displacement time graph.
• As velocity is the rate of change of position it is the gradient of a displacement time graph
• Instantaneous acceleration is calculated from a velocity time graph.
• As acceleration is defined as the rate of change of velocity, it is found as the gradient of a velocity time graph.
• Motion Graphs
• Plot the following displacement-time data on a graph.
• Take suitable measurements to construct the corresponding velocity-time and acceleration-time graphs
s /m 20.00 33.75 45.00 53.75 60.00 63.75 65.00 63.75 60.00 53.75 45.00 33.75 20.00 3.75 t /s 0 1 2 3 4 5 6 7 8 9 10 11 12 13
• Uniform Acceleration
• In the special case where the acceleration on an object is constant a number of equations can be derived to describe the object's motion.
• Consider the velocity-time graph for this object.
• Let the object change velocity from (t 1 ,u) to (t 2 ,v)
v t (t 1 ,u) (t 2 ,v)
• Uniform Acceleration
• By definition:
• The acceleration is the rate of change of velocity.
v t (t 1 ,u) (t 2 ,v)
• Uniform Acceleration
• By definition:
• The average velocity is the total displacement over time taken
v t (t 1 ,u) (t 2 ,v)
• Uniform Acceleration
• By definition:
• The total displacement is the area under the curve.
v t (t 1 ,u) (t 2 ,v)
• Uniform Acceleration
• By definition:
• The total displacement is the area under the curve.
v t (t 1 ,u) (t 2 ,v)
• Uniform Acceleration
• By eliminating t from equations 1 and 2:
v t (t 1 ,u) (t 2 ,v)
• Uniform Acceleration - Summary
• The 5 equations that can be used when a is constant are:
v t (t 1 ,u) (t 2 ,v)
• Practice
• An car has initial velocity 20ms -1 . It accelerates for 4.67s at a rate of 2.79ms -2 . What is its new velocity and what distance was covered whilst it accelerated?
• A ball is rolled across a rough table and covers a distance of 2.37m in 16.3s before coming to rest. What is the acceleration on the ball due to the friction (assume constant) and what was the ball's initial speed?
• Practice
• A cannon is fired horizontally. The shell has a constant deceleration of 0.05ms -2 and travels a distance of 1370m before hitting its target with speed 243ms -1 . How long is the cannon ball in the air and with what speed did it leave the barrel of the gun. (ignore gravity)
• A ball is kicked across smooth ground at a wall 15m away with speed 20ms -1 . The ball rebounds with speed 16ms -1 . The ball squashes by 20mm when in contact with the wall. Calculate the average acceleration exerted by the wall on the ball.
• An object falling freely (no forces acting except its weight) in a vacuum close to the Earth's surface will experience a downward acceleration of g =9.81ms -2 .
• This is the same value as the Earth's gravitational field strength and is due to the gradient of the gravitational field at this point.
• In Physics we approximate any small, smooth object falling in air as having an acceleration of 9.81ms -2
Free Fall
• Practice
• A ball is dropped from rest at a height of 3m. Calculate the velocity of the ball just before it strikes the ground and the time taken to fall.
• A ball is thrown vertically upwards with an initial speed of 2.5ms -1 . Calculate the maximum height of the ball and the total time of flight.
• A 2kg ball rolls down a smooth slope that is 5m long. Calculate the speed of the ball as it reaches the bottom of the slope.
• Terminal Velocity
• For real objects falling in viscous fluids the object suffers a resistive force that is (usually) proportional to the objects speed relative to the fluid.
• This means that as the object travels faster, the resistive forces also increase.
• At some point the resistive forces will be equal in magnitude to the weight force acting on the object.
• There will be no nett downwards force.
• There will be no downwards acceleration.
• There will be no increase in velocity.
• This maximum speed is called an object's terminal speed.
• Relative Velocity
• Imagine two objects (A and B) that are both moving.
• What does the velocity of A look like compared to the Universe?
• What does the same velocity look like when observed from B?
• The motion of two objects relative to each other is often of great interest to Physicists as this is how we observe the Universe in reality.
• We are never entirely stationary so we need to be able to calculate relative velocities.
• Relative Velocity
• The velocity of A as observed from B (the velocity of A relative to B (A/B)) is defined as:
• Here O represents some fixed co-ordinate system that other measurements can be made against.
• Be careful with calculations: v is a vector so direction must be accounted for!
• Practice
• Car A is travelling with velocity 70ms -1 due north.
• Car B is travelling with velocity 35ms -1 due north.
• Car C is travelling with velocity 50ms -1 due South.
• Car D is travelling with velocity 45ms -1 30 o South of East.
• Calculate:
• The velocity of A relative to B
• The velocity of A relative to C
• The velocity of B relative to C
• The velocity of C relative to B
• The velocity of D relative to C
• The velocity of A relative to D