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Instantaneous and Average Velocities In this example, we highlight how the average velocity approaches the instantaneous velocity as the time interval separating two events gets smaller. A toy remote-controlled car moves in a straight line along the x-axis such that its position (in metres) is given by x(t) 5 (0.250t3) m, where t is measured in seconds and the coefficient 0.250 has units of m/s3. Plot the car’s position versus time graph for the interval t1 = 1.00 s to t2 = 2.00 s. Find the average velocity of the car during the interval t1 = 1.00 s to t2 = 2.00 s, and include the corresponding line on your graph. Find the car’s average velocity during the interval t3 = 1.25 s to t4 = 1.75 s, and include the corresponding line on your graph. Find the car’s instantaneous velocity at t5 = 1.50 s, and include the corresponding line on your graph. Comment on the difference between average and instantaneous velocities as ∆t gets smaller.
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Polymer Physics
Can You Have a Non-zero Average Acceleration while Moving at a Constant Speed? A car races down the highway at a speed of 110. km/h, the highway loops around the base of a mountain, and 6.00 min later the car is moving at the same speed but now in the opposite direction. Calculate the average acceleration of the car.
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Polymer Physics
Using Derivatives to Calculate Velocity and Acceleration The position in metres (as a function of time, in seconds) for a particle moving along the x-axis is given by x(t) 5 = - 0.500t4 + 2.50t3 - 7.00t + 3.00. Find (a) the instantaneous velocity of the particle at t1 = 2.00 s (b) the instantaneous acceleration of the particle at t1 = 2.00 s (c) the average acceleration of the particle between t1 = 2.00 s and t2 = 3.00 s (d) the instantaneous acceleration of the particle at t3 = 2.50 s (e) the maximum speed the particle reaches in the first 5.00 s of its motion
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Polymer Physics
Braking time A car is moving at 92 km/h when the driver realizes that there is a traffic jam 80. m ahead, and the highway is blocked. Assuming that the average reaction time for a driver is 0.75 s, determine the following. (a) What is the minimum acceleration (in absolute value) the car must undergo to come to a stop without colliding with the cars ahead? (b) How much time does it take for the car to stop?
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Polymer Physics
A Graph-Matching Game: Using a Motion Detector to Analyze Motion A motion detector (or sonic ranger) is a device that uses ultrasound to measure the position of objects (Figure 3-20(a)). If an object is moving along a straight line, then a single motion detector is sufficient to determine its location. A motion detector is a “digital bat,” as it uses echolocation to determine where objects are in space and then deduce how they are moving. Echolocation uses the time it takes a sound pulse emitted from a detector to reflect off an object and return to the detector to determine the distance from the object to the detector. Since a motion detector can detect sound pulses many times a second, it can provide ample data about an object’s motion. Accompanying software, such as Logger Pro, can visualize these data points by plotting real-time x(t), vx(t), and ax(t) graphs. A student is asked to walk in front of the motion detector (Figure 3-20(a)) to match the given x(t) graph shown in Figu
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Polymer Physics
Launching Lunch A construction worker standing on a beam asks his very strong friend on the ground 9.00 m below to throw his lunch box up to him. The friend wants to throw the box straight up such that it has a velocity of 0.500 m/s up when it reaches the worker on the beam, to give him some leeway as he tries to catch it. What assumption should you make to solve the problem? At what speed should the friend throw the lunch box? How long will the lunch box be in the air before it is caught by the worker? How long does it take for the lunch box to cover the first 4.50 m of its trajectory? How high above its initial position will the lunch box be after half the total time of its motion? Plot the x(t), v_x(t), and a_x (t) graphs for the motion of the lunch box.
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Polymer Physics
A Horizontal Mass–Spring System A 2.00 kg mass, resting on a frictionless table, is attached to a spring of spring constant 20.0 N/m. The spring is stretched 20.0 cm by pulling on the mass, which is held stationary and then released. (a) Calculate the angular frequency, frequency, and period. (b) What are the amplitude and the phase constant of oscillations? (c) What are the maximum speed and maximum acceleration of the mass? (d) What is the displacement of the mass from equilibrium as a function of time?
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Polymer Physics
A Vertical Mass–Spring System A mass attached to a vertical spring is undergoing simple harmonic motion. A plot of the height of the mass measured with respect to the ground as a function of time is shown in Figure 13-19. The height of the mass at t=0 is 6.3 m. Find the height of the mass from the ground when the mass is at the equilibrium position. Find the amplitude and the period. Write an equation that describes the height of the mass with respect to the ground as a function of time.
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Polymer Physics
