More graphs of motion & Kinematic equations September 14-15, 2017

Area under Motion graphs Another important thing to know is that the area under the curve in a motion graph has meaning. area under V-T graph = Δx Δx = area under V-T graph How do you find the area of a triangle? Area = ½ base * height Δx = ½ * 5 s * 50 m/s = 125 m Area = Δx

Test your understanding Think about your answer, wait until I tell you, then show me the answer by holding up the correct number of fingers. . Which line has the highest acceleration? How do you know & how could you calculate that acceleration? Which line has the greatest displacement? How do you know & how could you calculate that displacement? 1 # 1 has the steepest slope. We could calculate a by finding the slope. # 2 has the greatest displacement. # 2 has the largest area under the line. We could calculate displacement by finding that area.

You do Use the graph of the motion of a toy train, below, to answer the questions. What is the acceleration between points C and D? How far does the train travel between points A and C? A = (40-20)/(6-5) = 20 m/s2 Find area: ½ (2)(20) = 20 m (2)(20) = 40 m total = 60 m

Kinematic Equations 𝒗 𝒂𝒗𝒈 = 𝒗𝒊+𝒗𝒇 𝟐 x = vi t + 𝟏 𝟐 𝒂𝒕2 vf2 = vi2 + 2ax Up until now, we’ve had just two kinematic equations: vavg = 𝜟𝒙 𝜟𝒕 and a = 𝜟𝒗 𝜟𝒕 These can be re-written in to more user-friendly forms … Let … t = the time for which the body accelerates a = acceleration vi = the velocity at time t = 0, the initial velocity vf = the velocity after time t, the final velocity x = the displacement covered in time t 𝒗 𝒂𝒗𝒈 = 𝒗𝒊+𝒗𝒇 𝟐 x = vi t + 𝟏 𝟐 𝒂𝒕2 vf2 = vi2 + 2ax vf = vi + at

Problem Solving Strategy Diagram the problem List known variables Determine what you are trying to find. Determine your strategy Solve the problem Evaluate your answer. Check whether the units, sign, and magnitude make sense.

Acceleration Problems – We do Grace is driving her sports car at 30 m/s when a ball rolls out into the street in front of her. Grace slams on the brakes and comes to a stop in 3.0 s. What was the acceleration of Grace’s car? Sketch: Known variables: What are we solving for: Strategy: Solution: Evaluate answer:

Acceleration Problems – We do Grace is driving her sports car at 30 m/s when a ball rolls out into the street in front of her. Grace slams on the brakes and comes to a stop in 3.0 s. What was the acceleration of Grace’s car? Sketch: Known variables: Vi = 30m/s Vf = 0 m/s t = 3.0 s What are we solving for: a Strategy: Use to solve for a. Solution: a = (vf – vi)/ t a = (0 m/s-30 m/s) / 3.0 s = -10 m/s2 Evaluate answer: Yes! Units are correct, sign makes sense, and magnitude is reasonable. vf = vi + at

Acceleration Problems – We do What is the displacement after 10.0 s of a mass whose initial velocity is 2.00 m/s and moves with acceleration a = 4.00 m/s2 ? Sketch: Known variables: What are we solving for: Strategy: Solution: Evaluate answer:

Acceleration Problems – We do What is the displacement after 10.0 s of a mass whose initial velocity is 2.00 m/s and moves with acceleration a = 4.00 m/s2 ? Sketch: Known variables: t = 10.0 s, vi = 2.00 m/s, a = 4.00 m/s2 What are we solving for: x Strategy: Solution: 220 m Evaluate answer: Yes! Distance and sign make sense. x = vi t + 𝟏 𝟐 𝒂𝒕2

Acceleration Problems – You do A car has an initial velocity of 5.0 m/s . When its displacement increases by 20.0 m, its velocity becomes 7.0 m/s. What is the acceleration? The car accelerate from rest to 28 m/s in 9.0 s. What distance does it travel? A Jet plane lands with a speed of 100 m/s and can accelerate uniformly at a maximum rate of -5.0 m/s2 as it comes to a rest. Can this plane land at an airport where the runway is 0.80 km long?

Acceleration Problems – You do A car has an initial velocity of 5.0 m/s . When its displacement increases by 20.0 m, its velocity becomes 7.0 m/s. What is the acceleration? Strategy – use Answer: 0.6 m/s2 The car accelerates from rest to 28 m/s in 9.0 s. What distance does it travel? Strategy, find a then find x. Answer: 126 m A Jet plane lands with a speed of 100 m/s and can accelerate uniformly at a maximum rate of -5.0 m/s2 as it comes to a rest. Can this plane land at an airport where the runway is 0.80 km long? Use Answer: No – it needs 1000 m vf2 = vi2 + 2ax vf2 = vi2 + 2ax