University Physics: Mechanics

University Physics: Mechanics Ch2. STRAIGHT LINE MOTION Lecture 2 Dr.-Ing. Erwin Sitompul http://zitompul.wordpress.com 2013

Solution for Homework 1: Truck You drives a truck along a straight road for 8.4 km at 70 km/h, at which point the truck runs out of gasoline and stops. Over the next 30 min, you walk another 2.0 km farther along the road to a gasoline station. (a) What is your overall displacement from the beginning of your drive to your arrival at the station? (b) What is the time interval Δt from the beginning of your drive to your arrival at the station?

Solution for Homework 1: Truck (c) What is your average velocity vavg from the beginning of your drive to your arrival at the station? Find it both numerically and graphically.

Solution for Homework 1: Truck (d) Suppose that to pump the gasoline, pay for it, and walk back to the truck takes you another 45 min. What is your average speed from the beginning of your drive to you return to the truck with the gas?

Instantaneous Velocity and Speed • Instantaneous velocity (or simply velocity) is the average velocity over a very short period of time interval • Velocity v at any instant is the slope of the position-time curve. • Instantaneous speed (or simply speed) is the magnitude of velocity, that is, speed is velocity without any indication of direction • What is the car’s vavg?

Instantaneous Velocity instantaneous velocity v= slope of curve at any point

Example: Elevator Cab Figure (a) on the left is an x(t) plot for an elevator cab that is initially stationary, then moves upward (which we take to be the positive direction of x), and then stops. Plot v(t). • The slope of x(t), which is the v(t), is zero in the intervals from 0 to 1 s and from 9 s on  the cab is stationary in these intervals. • During the interval bc, the slope is nonzero constant  the cab moves with constant velocity. • Between 1 s and 3 s the cab begins to move, and between 8 s and 9 s it slows down  the velocity of the cab varies. • Given v(t), can you determine x(t) exactly?

Motion with Constant Velocity constant Taking at time t0 = 0 the position is at x0 c = x0 Motion with constant velocity on x-t graph

Position, Time, and Velocity Whatis the velocity a cyclist for each stage of this trip?

Acceleration • Average acceleration is the ratio of change in velocity to the time interval. • Instantaneous acceleration (or simply acceleration) is the derivative of the velocity with respect to time.

Acceleration • Compare the a(t) curve with the v(t) curve. • Each point on the a(t) curve shows the derivative (slope) of the v(t) curve at the corresponding time. • When v is constant, the derivative is zero  a is also zero. • When the cab first begin to move, the v(t) curve has a positive derivation, which means that a(t) is positive. • When the cab slows to a stop, the derivative of v(t) is negative; that is, a(t) is negative.

Motion with Constant Acceleration constant Taking at time t0 = 0 the velocity equals v0 c = v0 Motions with constant acceleration on v-t graph

Motion with Constant Acceleration Taking at time t0 = 0 the position is at x0 c = x0 Motions with constant acceleration on v-t graph

Motion with Constant Acceleration From these two equations, the following equations can be derived:

Questions Whatis velocity in intervals A, B, C, D What is acceleration in intervals A, B, C 1 m/s 0 m/s2 2 m/s2 –1.5 m/s 0 m/s –0.5 m/s2 2 m/s

Example: Porsche Spotting a police car, you brake a Porsche from a speed of 100 km/h to a speed of 80.0 km/h during a displacement of 88.0 m, at a constant acceleration. (a) What is that acceleration?

Example: Porsche Spotting a police car, you brake a Porsche from a speed of 100 km/h to a speed of 80.0 km/h during a displacement of 88.0 m, at a constant acceleration. (b) How much time is required for the given decrease in speed?

Example: Porsche Car accelerating and decelerating

a > 0, a = 0, a < 0

Illustration: Overtaking Maneuver • Velocity vs. Time • Position vs. Time • Can you determine the exact value of vblue car? • Both cars move with constant velocity • Red car starts moving 4 seconds after the blue car • Both car meet at t = 7 s.

Trivia: How to Stop? You are driving a car with ever changing velocity but constant speed of 2 m/s. On your right is a steep cliff with the height of 50 cm. Directly in front of you there is a horse and behind you an elephant, both of which travel at your own speed. On your left there is a fire truck blocking you. How do you stop your car? Solution: Simple. Just ask the merry-go-round operator to stop!

Questions What are (a) Initial direction of travel? (b) Final direction of travel? (c) Does the particle stop momentarily? (d) Is the acceleration positive or negative? (e) Is the acceleration constant or varying? Which time periods indicates that an object moves at constant speed? – + Yes + Constant E, where a = 0

Questions How far does the runner travel in 16 s?

Example: Race A caravan moves with a constant velocity of 60 km/h along a straight road when it passes a roadster which is at rest. Exactly when the caravan passes the roadster, the roadster starts to move with an acceleration of 4 m/s2. (a) How much time does the roadster need to catch up the caravan? Thus, the roadster will catch up the caravan after 8.33 s.

Example: Race A caravan moves with a constant velocity of 60 km/h along a straight road when it passes a roadster which is at rest. Exactly when the caravan passes the roadster, the roadster starts to move with an acceleration of 4 m/s2. Distance traveled (b) How far does the roadster already move when it catches up the caravan? Both the vehicles travel 138.9 m before they pass each other again.

Example: Particle’s Movement The position of a particle is given by x = 4t2 – 2t + 10, where x is the distance from origin in meters and t the time in seconds. (a) Find the displacement of the particle for the time interval from t = 1 s to t = 2 s. (b) Find also the average velocity for the above given time interval. . (c) Find the instantaneous velocity of the particle at t = 0.5 s. At t = 0.5 s,

Homework 2: Aprilia vs. Kawasaki An Aprilia and a Kawasaki are separated by 200 m when they start to move towards each other at t = 0. 200 m The Aprilia moves with initial velocity 5 m/s and acceleration 4 m/s2. The Kawasaki runs with initial velocity 10 m/s and acceleration 6 m/s2. (a) Determine the point where the two motorcycles meet each other. (b) Determine the velocity of Aprilia and Kawasaki by the time they meet each other.

Homework 2A: Running Exercise You come late to a running exercise and your friends already run 200 m with constant speed of 4 m/s. The athletic trainer orders you to catch up your friends within 1 minute. If you run with constant speed, determine the minimum speed you have to take so that you can fulfill the trainer’s order. (b) If you run with minimum speed, determine the point where you catch up your friends.

Homework 2B: Runway Length An electron moving along the x axis has a position given by x = 16te–t m, where t is in seconds. How far is the electron from the origin when it momentarily stops? An airplane lands at a speed of 160 mi/h and decelerates at the rate of 10 mi/h/s. If the plane travels at a constant speed of 160 mi/h for 1.0 s after landing before applying the brakes, what is the total displacement of the aircraft between touchdown on the runway and coming to rest?

University Physics: Mechanics