SCI 101: Assignment 1, solved step by step

Answer True

1. Magnitude = square root of the sum of squared components: |v| = √(x² + y²).
2. |v| = √(3² + 4²) = √(9 + 16) = √25 = 5.

Watch out: 3-4-5 is a famous right triangle. Spot it and you skip the calculator.

Answer False

1. |v| = √(2² + 5²) = √(4 + 25) = √29 ≈ 5.39.
2. 5.39 is not 5, so the statement is False.

Watch out: Don't just read off the biggest component. Always square and add.

Answer A. 9.3

1. In 3D: |v| = √(x² + y² + z²).
2. Components: 2, 1, −9. Squares: 4, 1, 81.
3. |v| = √(4 + 1 + 81) = √86 ≈ 9.3.

Watch out: A magnitude is never negative (so D is out), and the square root of a sum of squares can't be a plain sum like 36.

Answer A. √38 m/s

1. Speed is the magnitude of the velocity vector.
2. Speed = √(2² + 3² + (−5)²) = √(4 + 9 + 25) = √38 m/s (≈ 6.16 m/s).

Watch out: Velocity is a vector; speed is its length. The minus sign disappears once squared.

Answer True

1. The three numbers are the x, y, z components.
2. Each component multiplies its unit vector: x goes with i, y with j, z with k.
3. So (5, 2, 3) = 5i + 2j + 3k. True.

Answer True

1. A vector A can be split along the axes: A = A_xi + A_yj + A_zk.
2. This is exactly how we resolve a vector into components, so the statement is True.

Answer A. (24, 30)

1. Multiplying a vector by a number multiplies every component by that number.
2. 6u = (6×4, 6×5) = (24, 30).

Watch out: Scalar multiplication never adds. Options B and D come from adding or only scaling one component.

Answer A. 4/5

1. Set the magnitude equal to 1: √(0² + (3/5)² + x²) = 1.
2. Square both sides: 9/25 + x² = 1.
3. x² = 1 − 9/25 = 16/25, so |x| = 4/5.

Watch out: 3-4-5 again: 3/5 and 4/5 are the sides of a unit-length hypotenuse.

Answer A. −u is a unit vector with opposite direction to u

1. The minus sign flips the direction but does not change the length.
2. |−u| = |u| = 1, so −u is still a unit vector, pointing the opposite way.

Answer A. A vector with the same magnitude but opposite direction

1. The negative of A is −A: same length, reversed direction.
2. Check: A + (−A) = 0, the zero vector.

Answer A. opposite

1. Same magnitude and same direction = equal vectors.
2. Same magnitude and reversed direction = opposite vectors.
3. Perpendicular means 90° apart; parallel means same or opposite line without needing equal length.

Watch out: The question says “opposite direction”, so the name is simply “opposite”.

Answer False

1. Equal vectors need both the same magnitude and the same direction.
2. Example: (3, 0) and (0, 3) both have magnitude 3, but they point in different directions. False.

Answer False

1. Take i and j, both unit vectors. Their sum is (1, 1) with magnitude √2 ≈ 1.41, not 1.
2. One counter-example is enough to make the statement False. (It would only work for the special angle of 120° between them.)

Answer A. Both their magnitudes and the angle between them

1. Resultant magnitude: R = √(A² + B² + 2AB cos θ).
2. The formula uses both lengths (A, B) and the angle θ, so all three matter.

Answer True

1. Equilibrium means no net force: ΣF = 0.
2. So the vector sum of the forces must be zero. True.

Answer A. 4

1. Multiply matching components, then add: u·v = u_xv_x + u_yv_y + u_zv_z.
2. = (2)(1) + (−3)(6) + (5)(4) = 2 − 18 + 20 = 4.

Watch out: The dot product gives a single number (a scalar), not a vector.

Answer False

1. Dot product = a number: A·B = AB cos θ. It is a scalar.
2. The cross product is the one that gives a vector. So the statement is False.

Answer A. their dot product is zero

1. A·B = AB cos θ. At θ = 90°, cos 90° = 0, so the dot product is 0.
2. A zero cross product means the vectors are parallel (sin θ = 0), the opposite situation.

Watch out: Dot = 0 → perpendicular. Cross = 0 → parallel.

Answer True

1. Perpendicular → dot product = 0.
2. (12)(3) + (x)(6) = 0 → 36 + 6x = 0.
3. 6x = −36 → x = −6. True.

Answer A. (4, 2)

1. Test each option: the perpendicular one has dot product 0 with (2, −4).
2. (4, 2): 2(4) + (−4)(2) = 8 − 8 = 0 ✓
3. (−2, 4): −4 − 16 = −20  |  (−2, −4): −4 + 16 = 12  |  (−4, 2): −8 − 8 = −16.

Watch out: Shortcut in 2D: swap the components and flip one sign. (2, −4) → (4, 2).

Answer A. (−10, −4, 15)

1. Formula: A×B = (A_yB_z − A_zB_y,  A_zB_x − A_xB_z,  A_xB_y − A_yB_x).
2. x: (0)(0) − (2)(5) = −10
3. y: (2)(−2) − (3)(0) = −4
4. z: (3)(5) − (0)(−2) = 15
5. Result: (−10, −4, 15).

Watch out: The y-component is the one people get wrong. Notice it is Az·Bx − Ax·Bz (the order is flipped).

Answer A. 40

1. Magnitude of a cross product: |v×u| = |v||u| sin θ.
2. = (10)(8) sin 30° = 80 × 0.5 = 40.

Watch out: Memorise sin 30° = 1/2. The dot product would use cos instead.

Answer False

1. It is exactly the other way around.
2. Dot product: A·B = B·A (commutative).
3. Cross product: A×B = −(B×A) (anti-commutative, the order flips the direction). So the statement is False.

Answer A. Instantaneous Velocity

1. Velocity at one specific moment is the instantaneous velocity, the limit of Δx/Δt as Δt → 0.
2. “Instantaneous speed” is only its magnitude, but the question asks about velocity.

Answer False

1. A speedometer shows how fast the car is going right now: the instantaneous speed.
2. Average speed = total distance ÷ total time, which a speedometer does not show. False.

Answer True

1. Velocity has magnitude and direction.
2. Same speed (constant magnitude) + straight line (constant direction) = constant velocity. True.

Answer True

1. Constant velocity means constant speed and no change in direction.
2. If the speed never changes, the average speed over any interval equals the speed at every instant. True.

Answer False

1. Velocity is a vector, so a change in direction is a change in velocity, even if the speed stays the same.
2. The U-turn reverses the direction, so velocity did not remain constant. False.

Answer True

1. Acceleration is the rate of change of velocity: a = Δv/Δt.
2. If speed increases, velocity changes, so a ≠ 0. True.

Answer False

1. Speeding up means velocity and acceleration point the same way.
2. The particle moves in the negative direction, so acceleration is negative, not positive. False.

Watch out: Speeding up: v and a same sign. Slowing down: opposite signs.

Answer False

1. Slowing down means acceleration points opposite to the velocity.
2. Velocity is negative, so acceleration is positive. The statement says negative, so it is False.

Answer A. The acceleration is zero

1. Constant speed in a straight line means the velocity does not change.
2. a = Δv/Δt = 0. The direction being negative does not matter. Acceleration is zero.

Answer True

1. Displacement is the straight-line vector from the start to the end position.
2. It starts 0.5 m above the ground and ends at the ground: 0.5 m, pointing down. True.

Answer True

1. Average velocity = displacement ÷ time = (x₂ − x₁)/Δt.
2. = (50 − 30)/4 = 20/4 = 5 m/s. True.

Answer True

1. Displacement = 40 − 20 = 20 m.
2. v_avg = 20/4 = 5 m/s. True.

Answer False

1. Displacement = 60 − 40 = 20 m.
2. v_avg = 20/4 = 5 m/s, not 4 m/s. False.

Answer A. 20.0 m/s

1. Δx = 30 − (−10) = 40 m (subtracting a negative adds).
2. Δt = 6 − 4 = 2 s.
3. v_avg = 40/2 = 20 m/s.

Watch out: Choice C (40) forgets to divide by the time interval.

Answer A. 24 m/s

1. Find the positions: x(3) = 3(9) − 1 = 26 m and x(5) = 3(25) − 1 = 74 m.
2. Δx = 74 − 26 = 48 m, Δt = 2 s.
3. v_avg = 48/2 = 24 m/s.

Watch out: This is average velocity, so use two positions. Don't differentiate (that would give the instantaneous velocity).

Answer A. 5 hours

1. At constant speed: d = vt, so t = d/v.
2. t = 250 km ÷ 50 km/h = 5 h.

Answer A. 20 m/s²

1. From a stop means v₀ = 0.
2. a = (v − v₀)/t = (60 − 0)/3 = 20 m/s².

Answer A. 36 m/s

1. Rearrange a = Δv/Δt into v = v₀ + aΔt.
2. Δt = 5 − 2 = 3 s.
3. v(5) = 6 + (10)(3) = 36 m/s.

Watch out: Don't use 5 s as the time. The acceleration acts only for the 3 s between the two instants.

Answer A. 4 sec

1. v = v₀ + at with v₀ = 0.
2. t = v/a = 16/4 = 4 s.

Answer A. 5 seconds

1. Use x = v₀t + ½at² with v₀ = 0.
2. 50 = ½(4)t² = 2t² → t² = 25.
3. t = 5 s.

Watch out: Options 25 and 50 forget to take the square root.

Answer A. 2 m

1. We don't know the time, so use the equation without time: v² = v₀² + 2as.
2. 6² = 4² + 2(5)s → 36 = 16 + 10s.
3. 10s = 20 → s = 2 m.

Watch out: Pick the equation that contains exactly the quantities you have and the one you want.

Answer A. Its acceleration is 9.8 m/s²

1. In free fall the only force is gravity, giving a constant acceleration g = 9.8 m/s² (downward).
2. Velocity keeps changing (B is wrong), acceleration isn't zero (C is wrong), and mass doesn't matter (D is wrong).

Answer True

1. Near the Earth's surface, g stays essentially constant at 9.8 m/s².
2. So free fall is motion with constant acceleration. True.

Answer False

1. Without air resistance every object falls with the same acceleration g = 9.8 m/s².
2. Heavier objects feel a larger force, but they also have more inertia, so the effects cancel. False.

Answer A. Air resistance has a greater effect on the feather than on the stone.

1. In a vacuum both would land together, because g is the same for all objects.
2. In air, drag matters much more for the light, wide feather than for the dense stone.
3. So the delay is caused by air resistance.

Watch out: Mass and a smaller gravitational force are true statements about the feather, but they are not why it falls slower.

Answer True

1. Going up, the velocity decreases. At the top it reaches 0 for an instant, then reverses.
2. So v = 0 at the maximum height. True.

Answer False

1. Velocity is zero at the top, but acceleration is not.
2. Gravity still acts, so a = 9.8 m/s² downward the whole time. False.

Watch out: Zero velocity does not mean zero acceleration. This is the most common free-fall mistake.

Answer A. 2.55 s

1. At the top, v = 0. Use v = v₀ − gt.
2. 0 = 25 − 9.8t → t = 25/9.8 = 2.55 s.

Answer False

1. At the top v = 0: t = v₀/g = 20/9.8 ≈ 2.04 s.
2. 2.04 s is not 1.5 s, so the statement is False.

Answer A. 15.1 m/s

1. We know the height, not the time, so use v² = v₀² − 2gh (gravity opposes the upward motion).
2. v² = 22² − 2(9.8)(13) = 484 − 254.8 = 229.2.
3. v = √229.2 ≈ 15.1 m/s.

Watch out: Zero would be the answer only at the maximum height, which here is 484/19.6 ≈ 24.7 m. The stone is still below that.

Answer A. v = 48.5 m/s downward

1. No initial speed is given, so treat the ball as released from rest: v₀ = 0.
2. v² = v₀² + 2gh = 0 + 2(9.8)(120) = 2352.
3. v = √2352 ≈ 48.5 m/s, downward.

Watch out: When a problem leaves out the starting speed, the intended answer assumes it starts from rest.

Answer A. 44.1 m

1. Dropped from rest: v₀ = 0, so y = ½gt².
2. y = ½(9.8)(3)² = 4.9 × 9 = 44.1 m.

Watch out: The 500 m is a distraction. The stone is nowhere near the ground after 3 s.