2. 1D Motion / Kinematics

Figure 2–42 shows the velocity of a train as a function of time. At what time was its velocity greatest?

High-speed motion pictures ($3500$ frames/second) of a jumping, $210–μg$ flea yielded the data used to plot the graph in Fig. E$2.54$. (See 'The Flying Leap of the Flea' by M. Rothschild, Y. Schlein, K. Parker, C. Neville, and S. Sternberg in the November $1973$ Scientific American.) This flea was about $2$ mm long and jumped at a nearly vertical takeoff angle. Use the graph to answer this question: Find the flea's acceleration at $0.5$ ms, $1.0$ ms, and $1.5$ ms.

High-speed motion pictures ($3500$ frames/second) of a jumping, $210–μg$ flea yielded the data used to plot the graph in Fig. E$2.54$. (See 'The Flying Leap of the Flea' by M. Rothschild, Y. Schlein, K. Parker, C. Neville, and S. Sternberg in the November $1973$ Scientific American.) This flea was about $2$ mm long and jumped at a nearly vertical takeoff angle. Use the graph to answer this question: Find the maximum height the flea reached in the first $2.5$ ms.

High-speed motion pictures ($3500$ frames/second) of a jumping, $210\text{–}\mu g$ flea yielded the data used to plot the graph in Fig. E$2.54$. (See 'The Flying Leap of the Flea' by M. Rothschild, Y. Schlein, K. Parker, C. Neville, and S. Sternberg in the November $1973$ Scientific American.) This flea was about $2$ mm long and jumped at a nearly vertical takeoff angle. Use the graph to answer this question: Is the acceleration of the flea ever zero? If so, when? Justify your answer.

The table shows test data for the Bugatti Veyron Super Sport, the fastest street car made. The car is moving in a straight line (the $x$-axis).

(a) Sketch a $v_x$-$t$ graph of this car's velocity (in mi/h) as a function of time. Is its acceleration constant?

(b) Calculate the car's average acceleration (in m/s2) between (i) $0$ and $2.1$ s; (ii) $2.1$ s and $20.0$ s; (iii) $20.0$ s and $53$ s. Are these results consistent with your graph in part (a)? (Before you decide to buy this car, it might be helpful to know that only $300$ will be built, it runs out of gas in $12$ minutes at top speed, and it costs more than $\$1.5$ million!)

A cat walks in a straight line, which we shall call the $x$-axis, with the positive direction to the right. As an observant physicist, you make measurements of this cat's motion and construct a graph of the feline's velocity as a function of time (Fig. E$2.30$). What distance does the cat move during the first $4.5$ s? From $t=0$ to $t=7.5$ s?

A cat walks in a straight line, which we shall call the $x$-axis, with the positive direction to the right. As an observant physicist, you make measurements of this cat's motion and construct a graph of the feline's velocity as a function of time (Fig. E$2.30$). What is the cat's acceleration at $t=3.0$ s? At $t=6.0$ s? At $t=7.0$ s?

A ball moves in a straight line (the $x$-axis). The graph in Fig. E$2.9$ shows this ball's velocity as a function of time. What are the ball's average speed and average velocity during the first $3.0$ s?

FIGURE EX2.8 showed the velocity graph of blood in the aorta. What is the blood's acceleration during each phase of the motion, speeding up and slowing down?

FIGURE EX2.9 shows the velocity graph of a particle. Draw the particle's acceleration graph for the interval 0 s ≤ t ≤ 4 s.

FIGURE EX2.12 shows the velocity-versus-time graph for a particle moving along the x-axis. Its initial position is at x₀ = 2 m at t₀ = 0 s. What are the particle's position, velocity, and acceleration at t = 3.0 s?

FIGURE EX2.12 shows the velocity-versus-time graph for a particle moving along the x-axis. Its initial position is at x₀ = 2 m at t₀ = 0 s. What are the particle's position, velocity, and acceleration at t = 1.0 s?

FIGURE EX2.32 shows the acceleration graph for a particle that starts from rest at t = 0 s. What is the particle's velocity at t = 6 s?

(II) A sports car accelerates approximately as shown in the velocity–time graph of Fig. 2–43. (The short flat spots in the curve represent manual shifting of the gears.) Estimate the car's average acceleration in second gear.

Given v(t) = 25 + 18t, where v is in m/s and t is in s, use calculus to determine the total displacement from t₁ = 1.3 s to t₂ = 3.6 s.