Adding vectors and relative velocity: Class 11 practice
Perpendicular vectors combine by Pythagoras: R = √(A² + B²), direction tan θ = B/A. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Pythagoras
- tan ratio
The idea behind adding vectors and relative velocity
Perpendicular vectors combine by Pythagoras: R = √(A² + B²), direction tan θ = B/A. River problems: the boat's ground velocity is the vector sum of its velocity in water and the current.
Step-by-step method
- Draw the vectors head to tail.
- For perpendicular vectors use R = √(A² + B²).
- Direction from tan θ.
Worked example 1 (Basic)
Find the resultant of 6 N east and 8 N north.
- R = √(6² + 8²) = 10 N
- θ = tan⁻¹(8/6) = 53.13° north of east
- Answer: 10 N at 53.13° N of E
Worked example 2 (Standard)
A boat heads straight across a 193 m river at 9 m/s; the current is 12 m/s. How long does it take and how far downstream does it land?
- t = 193/9 = 21.44 s
- Drift = 12 × 21.44 = 257.33 m
- Answer: 21.44 s, 257.33 m downstream
Worked example 3 (Challenge)
Rain falls vertically at 8 m/s; a cyclist rides at 15 m/s. At what angle to the vertical should she tilt her umbrella?
- Relative velocity of rain: 8 down, 15 towards her
- tan θ = 15/8 → θ = 61.93°
- Answer: 61.93° from vertical (forward)
Common mistakes and misconceptions
- Adding magnitudes directly
- Measuring the angle from the wrong axis
Practice questions
- (Basic) Find the resultant of 9 N east and 12 N north.
- (Basic) Find the resultant of 5 N east and 12 N north.
- (Standard) A boat heads straight across a 348 m river at 9 m/s; the current is 12 m/s. How long does it take and how far downstream does it land?
- (Standard) A boat heads straight across a 139 m river at 6 m/s; the current is 8 m/s. How long does it take and how far downstream does it land?
- (Challenge) Rain falls vertically at 5 m/s; a cyclist rides at 12 m/s. At what angle to the vertical should she tilt her umbrella?
- (Challenge) Rain falls vertically at 6 m/s; a cyclist rides at 8 m/s. At what angle to the vertical should she tilt her umbrella?
Answer key with full working
- Q1: R = √(9² + 12²) = 15 N → θ = tan⁻¹(12/9) = 53.13° north of east → Answer: 15 N at 53.13° N of E
- Q2: R = √(5² + 12²) = 13 N → θ = tan⁻¹(12/5) = 67.38° north of east → Answer: 13 N at 67.38° N of E
- Q3: t = 348/9 = 38.67 s → Drift = 12 × 38.67 = 464 m → Answer: 38.67 s, 464 m downstream
- Q4: t = 139/6 = 23.17 s → Drift = 8 × 23.17 = 185.33 m → Answer: 23.17 s, 185.33 m downstream
- Q5: Relative velocity of rain: 5 down, 12 towards her → tan θ = 12/5 → θ = 67.38° → Answer: 67.38° from vertical (forward)
- Q6: Relative velocity of rain: 6 down, 8 towards her → tan θ = 8/6 → θ = 53.13° → Answer: 53.13° from vertical (forward)
Syllabus note
This skill is listed under "Motion in a Plane" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is adding vectors and relative velocity in?
"Motion in a Plane", Class 11 Physics (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.