A particle moves along a straight line with velocity given by v = (6 – 3t) where v is in m/s and t in seconds. Determine when the particle returns to its starting point.
Displacement is \(s = \int v \, dt = 6t - 1.5t^2\). To return to the starting point, \(s = 0 \implies 6t - 1.5t^2 = 0 \implies t = 4\text{ s}\).
A car is moving with velocity of 20 m/s on a straight road. A scooterist wishes to overtake the car in 60 s. If the car is at a distance of 1.5 km ahead, then the velocity with which the scooterist has to chase the car is
Relative velocity required to cover 1500 m in 60 s is \(v_{\text{rel}} = \frac{1500}{60} = 25\text{ m/s}\). Since \(v_{\text{rel}} = v_s - v_c ⇒ v_s = v_c + v_{\text{rel}} = 20 + 25 = 45\text{ m/s}\).
Assertion (A): Path of a projected ball is parabolic in uniform gravitational field for oblique projection in absence of air resistance.
Reason (R): Gravitational force is always act perpendicular to velocity during the motion of a projectile.
The path of a projectile is a parabola under constant gravitational acceleration. Gravitational force acts vertically downwards, which is perpendicular to the velocity only at the highest point of its trajectory, so R is false.
Assertion (A): In any interval, the magnitude of displacement is always less than or equal to the distance travelled.
Reason (R): For a particle travelling in a straight line with constant acceleration, the magnitude of the change in the velocity during any interval is always less than or equal to the change in the speed during that interval.
Displacement is the straight-line distance, so its magnitude is always \(le\) distance. When a particle reverses its direction of motion, the magnitude of change in velocity can be greater than the change in speed, so R is false.
Assertion (A): A particle with constant acceleration always moves along a straight line.
Reason (R): A particle with constant acceleration will not change direction of motion.
A projectile experiences constant acceleration (g) but follows a parabolic path. Also, a ball thrown vertically upwards under gravity has constant acceleration but reverses its direction of motion, making both statements false.
Assertion (A): If initial velocity is negative and acceleration is positive then motion is retarded (initially).
Reason (R): If initial velocity is negative but acceleration is positive then displacement of a particle can never be positive.
When velocity and acceleration have opposite signs, the speed decreases (retardation). After stopping, the positive acceleration will move the particle in the positive direction, which can result in positive displacement, so R is false.
Assertion (A): Path of a projected ball is parabolic in uniform gravitational field for oblique projection in absence of air resistance.
Reason (R): Gravitational force is always act perpendicular to velocity during the motion of a projectile.
Assertion (A) is true because projectile motion under gravity without air resistance follows a parabolic path.
Reason (R) is false as gravitational force acts perpendicular to velocity only at the peak of the trajectory, not always. Thus, (A) is true, (R) is false.
Assertion (A): In any interval, the magnitude of displacement is always less than or equal to the distance travelled.
Reason (R): For a particle travelling in a straight line with constant acceleration, the magnitude of the change in the velocity during any interval is always less than or equal to the change in the speed during that interval.
Assertion (A) is true because displacement is the shortest path, so its magnitude is always less than or equal to the distance travelled. Reason (R) is false. For example, if velocity changes from \(-5 \text{ m/s}\) to \(+5 \text{ m/s}\), change in velocity magnitude is \(10 \text{ m/s}\), but change in speed is \(0 \text{ m/s}\). Thus, (A) is true, (R) is false.
Assertion (A): A particle with constant acceleration always moves along a straight line.
Reason (R): A particle with constant acceleration will not change direction of motion.
Assertion (A) is false. Projectile motion has constant acceleration (\(\vec{g}\)) but follows a parabolic path, not a straight line.
Reason (R) is false. A particle can change direction even with constant acceleration (e.g., projectile motion, or an object slowing down and reversing direction).
Thus, both (A) and (R) are false.
Assertion (A): If initial velocity is negative and acceleration is positive then motion is retarded (initially).
Reason (R): If initial velocity is negative but acceleration is positive then displacement of a particle can never be positive.
Assertion (A) is true because if velocity and acceleration have opposite signs (negative velocity, positive acceleration), the object is slowing down (retarding) initially.
Reason (R) is false. An object with negative initial velocity and positive acceleration can eventually reverse direction and achieve positive displacement (e.g., if it starts at \(x=0\), it will eventually cross \(x=0\) and move to positive \(x\)).
Thus, (A) is true, (R) is false.