Solving Word Problems - Kinematics

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Word Problems — solving constant-acceleration problems with the GUESS method

Warm-Up

An object has a negative velocity and a positive acceleration. Is it speeding up or slowing down?

Video Instruction

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C. Solving Word Problems

Ex 1

A car accelerates uniformly from rest at a rate of +10 m/s². How far will the car travel in 4 seconds?

Step 1 — Read and underline all the important information

Step 2 — Turn all your information into letters

Given

V_i = 0 (starts from rest) a = +10 m/s² t = 4 sec d = ?

↓

Step 3 — Substitute and Solve

Δd = V_it + ½aΔt²

= 0 m/s(4 sec) + ½(10 m/s²)(4 sec)²

Δd = 0 + 5(16)

Δd = 80 m

D. Deriving the 2nd Displacement Equation

Goal

V_f² = V_i² + 2aΔd

Start with

Δd = VΔt where V = (V_f + V_i)/2

Since V_f = V_i + aΔt, then Δt
= (V_f − V_i)/a

Δd = [(V_f + V_i)/2] · [(V_f − V_i)/a]

Cross multiply

2aΔd = V_f² − V_i²

Rearrange

V_f² = V_i² + 2aΔd

Wheelchair basketball player bouncing a ball

Ex 2

A car with an initial velocity of +6 m/s accelerates at a rate of 2 m/s². Find the car's final velocity after it travels 7 m from its initial position.

Step 1 — Read and underline all the important information

Step 2 — Turn all your information into letters

Given

V_i = 6 m/s a = 2 m/s² V_f = ? d = 7 m

↓

Step 3 — Substitute and Solve

V_f² = V_i² + 2aΔd

= (6 m/s)² + 2(2 m/s²)(7 m)

V_f² = 36 + 4(7) = 64

V_f = 8 m/s

Check Your Understanding

5 quick questions — pick an answer to see if you’re right.

1. A car accelerates uniformly from rest at +4 m/s². How far does it travel in 5 s?

2. A car moving at +3 m/s accelerates at 2 m/s² over 4 m. What is its final velocity?

3. An object has a negative velocity and a positive acceleration. What is it doing?

4. In a word problem, "starts from rest" tells you that...

5. Which equation lets you find final velocity WITHOUT knowing the time?

NYSSLS Regents Practice · Kinematics · Engineering Design

Designing a Highway On-Ramp

Traffic engineers are designing an on-ramp so that cars can safely merge onto a highway where traffic moves at about 31 m/s. A typical car enters the ramp at 12 m/s and accelerates uniformly at 2.5 m/s² along a straight ramp that is 150 m long.

What is the car’s speed at the end of the ramp?

▶ Want a challenge? Open 6 more Regents-style questions (harder · mixed formats)

Challenge score: 0 / 6

Select all that apply

C1. Place a check mark in three boxes to indicate the quantities that must be known to determine a car’s speed at the end of the ramp without knowing the time.

Multiple choice

C2. How much time does the car spend on the ramp?

Substitute values

C3. Substitute numerical values into a mathematical representation that can be used to determine the acceleration a car needs to reach exactly 31 m/s by the end of the 150-m ramp. Then calculate the acceleration.

a = (v_f = m/s)² − (v_i = m/s)²2( m) = m/s²

Support or Refute · Justify

C4. A planner makes the following claim.

Claim: Doubling the ramp length to 300 m would double the car’s speed at the end of the ramp to about 60 m/s.

Place a check mark in either the Support or Refute box. Justify your response using a mathematical relationship.

Support Refute

Multiple choice

C5. To save money, a planner suggests shortening the ramp to 100. m. Which evaluation is correct?

Constructed response

C6. The engineers are considering a ramp longer than 150 m. Identify one benefit and one drawback of this design change.

Displacement Plots →

Bonus

Ever see an object roll UP hill?