Kinematics

Question 1

In some appropriate units, time (t) and position (x) relation of a moving particle is given by t = x² + x. The acceleration of the particle is

Accepted Answer

−2/(2x + 1)³. Differentiating t = x² + x with respect to x gives dt/dx = 2x + 1, so the velocity is v = dx/dt = 1/(2x + 1). Acceleration then follows from a = v dv/dx = (2x + 1)⁻¹ x [−2(2x + 1)⁻²] = −2/(2x + 1)³.

Answer

−2/(x + 2)³

Answer

+2/(x + 1)³

Answer

2/(2x + 1)

Question 2

Two cities X and Y are connected by a regular bus service with a bus leaving in either direction every T min. A girl is driving scooty with a speed of 60 km/h in the direction X to Y notices that a bus goes past her every 30 minutes in the direction of her motion, and every 10 minutes in the opposite direction. Choose the correct option for the period T of the bus service and the speed (assumed constant) of the buses.

Accepted Answer

15 min, 120 km/h. Consecutive buses are always the same distance vT apart. A bus overtaking her closes that gap at v − 60, while an oncoming bus closes it at v + 60, so (v − 60) x 30 = (v + 60) x 10 = vT. The first equality gives v = 120 km/h, and substituting back gives T = 15 min.

Answer

9 min, 40 km/h

Answer

25 min, 100 km/h

Answer

10 min, 90 km/h

Question 3

Consider a particle moving along a straight line, whose position as a function of time is given by s(t) = αt² − βt + γ, where α = 1 m s⁻², β = 6 m s⁻¹ and γ = 5 m. The average speed of the particle, in m s⁻¹, from t = 0 to t = 6 s is:

Accepted Answer

3. Average SPEED uses distance travelled, not displacement, and that distinction is the whole question. The velocity v = 2t − 6 vanishes at t = 3 s, so the particle reverses: it covers 9 m out and 9 m back, 18 m in 6 s, giving 3 m s⁻¹. Its average velocity over the same interval is zero.

Answer

12

Answer

6

Answer

0

Q1.In some appropriate units, time (t) and position (x) relation of a moving particle is given by t = x² + x. The acceleration of the particle is

Q3.Consider a particle moving along a straight line, whose position as a function of time is given by s(t) = αt² − βt + γ, where α = 1 m s⁻², β = 6 m s⁻¹ and γ = 5 m. The average speed of the particle, in m s⁻¹, from t = 0 to t = 6 s is: