document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165645>Question 1165645</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An automobile accelerates from rest at 1.7 m/s2 for 15.5 s. The speed is then held constant for 23.4 s, after which there is an acceleration of -3.5 m/s2 until the automobile stops. What is the total distance traveled? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852954 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852954\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "# Given parameters<BR>\n" );
document.write( "a1 = 1.7  # m/s^2<BR>\n" );
document.write( "t1 = 15.5 # s<BR>\n" );
document.write( "v0_1 = 0  # m/s\r<BR>\n" );
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document.write( "a3 = -3.5 # m/s^2<BR>\n" );
document.write( "v_final_3 = 0 # m/s\r<BR>\n" );
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document.write( "t2 = 23.4 # s\r<BR>\n" );
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document.write( "# --- Phase 1: Acceleration ---<BR>\n" );
document.write( "# 1. Final velocity of Phase 1 (v_f1), which is the constant velocity for Phase 2 (v2)<BR>\n" );
document.write( "v_f1 = v0_1 + a1 * t1<BR>\n" );
document.write( "v2 = v_f1\r<BR>\n" );
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document.write( "# 2. Distance traveled in Phase 1 (d1)<BR>\n" );
document.write( "d1 = v0_1 * t1 + 0.5 * a1 * t1**2\r<BR>\n" );
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document.write( "# --- Phase 2: Constant Speed ---<BR>\n" );
document.write( "# 3. Distance traveled in Phase 2 (d2)<BR>\n" );
document.write( "d2 = v2 * t2\r<BR>\n" );
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document.write( "# --- Phase 3: Deceleration ---<BR>\n" );
document.write( "# 4. Initial velocity of Phase 3 (v_03) is v_f1<BR>\n" );
document.write( "v0_3 = v_f1<BR>\n" );
document.write( "# Distance traveled in Phase 3 (d3): v_f^2 = v_0^2 + 2*a*d<BR>\n" );
document.write( "# 0 = v_f1^2 + 2*a3*d3<BR>\n" );
document.write( "d3 = -(v_f1**2) / (2 * a3)\r<BR>\n" );
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document.write( "# --- Total Distance ---<BR>\n" );
document.write( "# 5. Total distance (D_total)<BR>\n" );
document.write( "D_total = d1 + d2 + d3\r<BR>\n" );
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document.write( "print(f\"Final velocity of Phase 1 (v_f1): {v_f1} m/s\")<BR>\n" );
document.write( "print(f\"Distance in Phase 1 (d1): {d1} m\")<BR>\n" );
document.write( "print(f\"Distance in Phase 2 (d2): {d2} m\")<BR>\n" );
document.write( "print(f\"Distance in Phase 3 (d3): {d3} m\")<BR>\n" );
document.write( "print(f\"Total Distance (D_total): {D_total} m\")\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Final velocity of Phase 1 (v_f1): 26.349999999999998 m/s<BR>\n" );
document.write( "Distance in Phase 1 (d1): 204.2125 m<BR>\n" );
document.write( "Distance in Phase 2 (d2): 616.5899999999999 m<BR>\n" );
document.write( "Distance in Phase 3 (d3): 99.18892857142855 m<BR>\n" );
document.write( "Total Distance (D_total): 919.9914285714284 m\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "The total distance traveled by the automobile is approximately **$920.0$ meters**.\r<BR>\n" );
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document.write( "The movement is calculated in three phases: acceleration, constant speed, and deceleration.\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "## 1\. Phase 1: Acceleration\r<BR>\n" );
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document.write( "The automobile accelerates from rest ($v_0 = 0 \text{ m/s}$) for $15.5 \text{ s}$.\r<BR>\n" );
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document.write( "### A. Final Speed ($v_1$)\r<BR>\n" );
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document.write( "This speed will be the constant speed for Phase 2.<BR>\n" );
document.write( "$$v_1 = v_0 + a_1 t_1$$<BR>\n" );
document.write( "$$v_1 = 0 + (1.7 \text{ m/s}^2)(15.5 \text{ s})$$<BR>\n" );
document.write( "$$\mathbf{v_1 = 26.35 \text{ m/s}}$$\r<BR>\n" );
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document.write( "### B. Distance Traveled ($d_1$)\r<BR>\n" );
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document.write( "$$d_1 = v_0 t_1 + \frac{1}{2} a_1 t_1^2$$<BR>\n" );
document.write( "$$d_1 = 0 + \frac{1}{2} (1.7 \text{ m/s}^2)(15.5 \text{ s})^2$$<BR>\n" );
document.write( "$$\mathbf{d_1 \approx 204.21 \text{ m}}$$\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "## 2\. Phase 2: Constant Speed\r<BR>\n" );
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document.write( "The speed is held constant ($v_2 = 26.35 \text{ m/s}$) for $23.4 \text{ s}$.\r<BR>\n" );
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document.write( "### A. Distance Traveled ($d_2$)\r<BR>\n" );
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document.write( "$$d_2 = v_2 t_2$$<BR>\n" );
document.write( "$$d_2 = (26.35 \text{ m/s})(23.4 \text{ s})$$<BR>\n" );
document.write( "$$\mathbf{d_2 \approx 616.59 \text{ m}}$$\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "## 3\. Phase 3: Deceleration\r<BR>\n" );
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document.write( "The automobile decelerates from $v_0 = 26.35 \text{ m/s}$ at $a_3 = -3.5 \text{ m/s}^2$ until it stops ($v_f = 0 \text{ m/s}$).\r<BR>\n" );
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document.write( "### A. Distance Traveled ($d_3$)\r<BR>\n" );
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document.write( "We use the time-independent kinematic equation:<BR>\n" );
document.write( "$$v_f^2 = v_0^2 + 2 a_3 d_3$$<BR>\n" );
document.write( "$$0^2 = (26.35 \text{ m/s})^2 + 2 (-3.5 \text{ m/s}^2) d_3$$<BR>\n" );
document.write( "$$0 = 694.3225 - 7.0 d_3$$<BR>\n" );
document.write( "$$7.0 d_3 = 694.3225$$<BR>\n" );
document.write( "$$\mathbf{d_3 \approx 99.19 \text{ m}}$$\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "## 4\. Total Distance Traveled\r<BR>\n" );
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document.write( "The total distance is the sum of the distances from all three phases:<BR>\n" );
document.write( "$$D_{total} = d_1 + d_2 + d_3$$<BR>\n" );
document.write( "$$D_{total} = 204.21 \text{ m} + 616.59 \text{ m} + 99.19 \text{ m}$$<BR>\n" );
document.write( "$$\mathbf{D_{total} \approx 919.99 \text{ m}}$$ </SPAN>\n" );
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