Question 2.4: The Daytona 500 GOAL Apply the basic kinematic equations. PR...
The Daytona 500
GOAL Apply the basic kinematic equations.
PROBLEM (a) A race car starting from rest accelerates at a constant rate of 5.00 m/s². What is the velocity of the car after it has traveled 1.00 × 10² ft? (b) How much time has elapsed? (c) Calculate the average velocity two different ways.
STRATEGY (a) We’ve read the problem, drawn the diagram in Figure 2.16, and chosen a coordinate system (steps 1 and 2). We’d like to find the velocity v after a certain known displacement Δx. The acceleration a is also known, as is the initial velocity v_0 (step 3, labeling, is complete), so the third equation in Table 2.4 looks most useful for solving part (a). Given the velocity, the first equation in Table 2.4 can then be used to find the time in part (b). Part (c) requires substitution into Equations 2.2 and 2.7, respectively.
Table 2.4 Equations for Motion in a Straight Line Under Constant Acceleration
| Equation | Information Given by Equation |
| v = v_0 + at | Velocity as a function of time |
| Δx = v_0 t + \frac{1}{2}at² | Displacement as a function of time |
| v² = v_0 ² + 2aΔx | Velocity as a function of displacement |
Note: Motion is along the x-axis. At t = 0, the velocity of the particle is v_0.
\bar{v} \equiv \frac{Δx}{Δt} = \frac{x_f – x_i}{t_f – t_i} (2.2)
\bar{v} = \frac{v_0 + v}{2} (for constant a) (2.7)
(a) Convert units of Δx to SI, using the information in the inside front cover.
1.00 × 10² ft = (1.00 × 10² ft)(\frac{1 m}{3.28 ft}) = 30.5 mWrite the kinematics equation for v² (step 4): v² = v_0 ² + 2a Δx
Solve for v, taking the positive square root because the car moves to the right (step 5):
v = \sqrt{v_0 ² + 2a Δx}Substitute v_0 = 0, a = 5.00 m/s², and Δx = 30.5 m:
v = \sqrt{v_0 ² + 2a Δx} = \sqrt{(0)² + 2(5.00 m/s²)(30.5 m)} = 17.5 m/s(b) How much time has elapsed?
Apply the first equation of Table 2.4: v = at + v_0
Substitute values and solve for time t:
17.5 m/s = (5.00 m/s²)tt = \frac{17.5 m/s}{5.00 m/s²} = 3.50 s
(c) Calculate the average velocity in two different ways.
Apply the definition of average velocity, Equation 2.2:
\bar{v} = \frac{x_f – x_i}{t_f – t_i} = \frac{30.5 m}{3.50 s} = 8.71 m/sApply the definition of average velocity in Equation 2.7:
\bar{v} = \frac{v_0 + v}{2} = \frac{0 + 17.5 m/s}{2} = 8.75 m/sREMARKS The answers are easy to check. An alternate technique is to use Δx = v_0t + \frac{1}{2} at² to find t and then use the equation v = v_0 + at to find v. Notice that the two different equations for calculating the average velocity, due to rounding, give slightly different answers.