AP Physics 1: Multi-Step Kinematics Problems

Mastering multi-step kinematics is what separates adequate physics students from excellent ones. On the AP Physics 1 exam, you will almost certainly face problems where an object’s motion changes character—speeding up, cruising, and then slowing down—all in one scenario. These problems test your true understanding of the kinematic equations by requiring you to deconstruct a complex journey into manageable segments, a skill vital for engineering analysis and understanding real-world motion.

The Foundational Strategy: Segment and Conquer

The absolute core strategy for solving any multi-step kinematics problem is the segment and conquer approach. You must mentally, or literally with your pencil, divide the object’s entire trip into distinct phases where the acceleration is constant (which can be zero). A common sequence is acceleration (changing velocity), followed by constant velocity (zero acceleration), and finally deceleration (acceleration opposite the direction of motion).

Your first task is to identify and label these segments. For example, a car starting from rest, accelerating onto a highway, cruising, and then braking to a stop has three clear segments: Segment 1 (acceleration), Segment 2 (constant velocity), and Segment 3 (deceleration). For each segment, you will create a separate little "data box" in your work area, listing the five kinematic variables: initial velocity ($v_i$), final velocity ($v_f$), acceleration ($a$), displacement ($\Delta x$), and time interval ($\Delta t$). Crucially, the final velocity of one segment becomes the initial velocity of the next—this is the golden thread that connects your solution.

Identifying and Using Transition Points

The transition points between segments are the linchpins of your solution. At the exact moment a car stops accelerating and begins moving at constant velocity, its speed is neither the initial nor the final speed of the first segment—it is both. It is the final velocity of the acceleration segment and the constant velocity for the entire next segment.

Let’s apply this with a concrete, step-by-step example. A skateboarder starts from rest and accelerates at $+2.0{\text{m/s}}^2$ for $3.0\text{seconds}$. She then rolls at constant velocity for $5.0\text{seconds}$ before hitting a rough patch, decelerating at $-1.5{\text{m/s}}^2$ until she stops. What is her total displacement?

• Segment 1 (Acceleration): We know $v_{i1}=0$, $a_1=+2.0{\text{m/s}}^2$, $\Delta t_1=3.0\text{s}$. We first find her velocity at the end of this segment using $v_f=v_i+a\Delta t$. $v_{f1}=0+(2.0)(3.0)=+6.0\text{m/s}$. We also find the displacement using $\Delta x=v_i\Delta t+\frac{1}{2}a(\Delta t{)}^2$. $\Delta x_1=0+\frac{1}{2}(2.0)(3.0{)}^2=+9.0\text{m}$.
• Segment 2 (Constant Velocity): The transition point gives us $v_{i2}=v_{f1}=+6.0\text{m/s}$. Acceleration $a_2=0$. Time $\Delta t_2=5.0\text{s}$. With constant velocity, displacement is simple: $\Delta x_2=v\Delta t=(6.0\text{m/s})(5.0\text{s})=+30.0\text{m}$. Her final velocity here is still $+6.0\text{m/s}$.
• Segment 3 (Deceleration): The transition gives $v_{i3}=+6.0\text{m/s}$. She comes to a stop, so $v_{f3}=0$. Acceleration $a_3=-1.5{\text{m/s}}^2$. We need displacement. The best equation is $v_f^2=v_i^2+2a\Delta x$. Rearranging: $\Delta x_3=(v_f^2-v_i^2)/(2a)=(0^2-(6.0{)}^2)/(2\times -1.5)=(-36)/(-3.0)=+12.0\text{m}$.
• Total: The total displacement is the sum of the segment displacements: $\Delta x_{\text{total}}=+9.0+30.0+12.0=+51.0\text{m}$.

Advanced Tactics: Working Backwards and Graphical Cross-Checks

Sometimes, problems give you the total displacement or total time and ask for an intermediate value, like the acceleration during one segment. This requires you to set up a system of equations. You write equations for the unknown variable in each segment, knowing they must sum to the given total. For instance, if total time ($T$) is known, then $\Delta t_1+\Delta t_2+\Delta t_3=T$. You may need to express $\Delta t$ for one segment in terms of velocities and acceleration using $v_f=v_i+a\Delta t$.

A powerful strategy for both solving and checking your work is graphical thinking. Sketching a rough velocity-vs.-time ($v$-$t$) graph immediately visualizes the segments: an upward slope (acceleration), a horizontal line (constant velocity), and a downward slope (deceleration). The displacement during any interval is the area under the v-t curve. The total displacement in our skateboarder problem is the sum of the areas of a triangle, a rectangle, and another triangle. This provides an excellent independent check of your algebraic work.

Furthermore, always be mindful of vector direction. In one-dimensional motion, you must define a positive direction (e.g., to the right is $+$) and consistently assign positive or negative signs to velocities and accelerations. A car slowing down has a velocity and an acceleration with opposite signs. If velocity is $+20\text{m/s}$ and acceleration is $-4{\text{m/s}}^2$, the car is decelerating.

Common Pitfalls

1. Reusing Initial Conditions: The most common error is using the overall initial velocity (often zero) for every segment. Remember: the final velocity of Segment 1 is the initial velocity for Segment 2. Always check your data box for each segment.
2. Sign Confusion in Deceleration: Students often incorrectly plug in a deceleration value as a positive number. If an object is slowing down in your positive direction, its acceleration is negative. Carefully assign signs based on your defined coordinate system and whether the object is speeding up or slowing down.
3. Mixing Segment Variables: Do not accidentally use the time from Segment 2 in a kinematic equation for Segment 1 unless you have explicitly linked them through an intermediate variable. Keep the data for each segment meticulously organized and separate until the final summation step.
4. Assuming Constant Velocity Means Zero Distance: An object moving at constant velocity is still displacing. Do not skip a segment just because acceleration is zero; its displacement ($\Delta x=v\Delta t$) is often a major component of the total journey.

Summary

• Segment and Conquer: Break the total motion into distinct phases where acceleration is constant. Analyze each segment independently using the kinematic equations.
• Master Transition Points: The final velocity of one segment is the initial velocity of the next. This is the critical link that allows you to solve the problem sequentially.
• Organize Relentlessly: Create a labeled data box ($v_i$, $v_f$, $a$, $\Delta x$, $\Delta t$) for each segment to avoid variable confusion and sign errors.
• Think Graphically: A quick $v$-$t$ graph clarifies the relationship between segments and provides an area-based method to verify your calculated displacements.
• Mind the Signs: Consistently define a positive direction. Deceleration is not a positive number; it is acceleration opposite the direction of velocity.
• Check Totals: Your sum of segment times or displacements must match any given totals in the problem, providing a final check on your work.