A particle has initial velocity \(2 \text{m/s}\) and has constant acceleration such that after \(2 \text{s}\) its velocity becomes \(3 \text{m/s}\). The distance travelled by particle in this duration is
Since acceleration is constant, the distance is given by \(s = \left(\frac{u + v}{2}\right) t = \left(\frac{2 + 3}{2}
ight) \times 2 = 5 \text{m}\).
A particle is moving in x-y plane such that its x and y coordinates changes with time according to relation, \(x = 3t^2\) & \(y = 5t\) (here x & y are in m & t is in s). Speed of the particle at \(t = 2\) s, will be
The velocity components are \(v_x = \frac{dx}{dt} = 6t\) and \(v_y = \frac{dy}{dt} = 5\). At \(t = 2\) s, \(v_x = 12\text{ m/s}\) and \(v_y = 5\text{ m/s}\). Speed is \(v = \sqrt{v_x^2 + v_y^2} = \sqrt{12^2 + 5^2} = 13\text{ m/s}\).
Given below are two statements one is labelled as assertion (A) and reason (R)
Assertion: The length of actual path travelled by a body in given time interval is always equal to displacement.
Reason: If displacement is zero, then body is either at rest or it has returned to initial position.
Choose the correct option.
Distance (actual path length) is greater than or equal to displacement magnitude, so the assertion is false. If displacement is zero, the body either remained at rest or returned to its starting point, making the reason true.
In the following question, a statement of Assertion (A) is followed by a statement of Reason (R).
Assertion (A): If two particles, moving along straight line with constant velocities have to meet, the relative velocity must be along the line joining the two particles.
Reason (R): Relative motion means motion of one particle as viewed from the other particle.
For two particles to meet, the relative velocity vector must align with the line joining them so that from one's frame, the other moves directly towards it.
Two particles are projected with same initial velocity one makes angle \(\theta\) with horizontal while other makes an angle \(\theta\) with vertical. If their common range is R then product of their time of flight is directly proportional to:
(1999)
Concept: Time of flight for complementary angles and range formula.
Formula: \(T = (2u sin \alpha) / g\), \(R = (u^2 sin 2\alpha) / g\).
For angles \(\theta\) and \(90° - \theta\), times are \(T_1 = (2u sin \theta) / g\) and \(T_2 = (2u cos \theta) / g\).
Their product \(T_1 T_2 = (4u^2 sin \theta cos \theta) / g^2 = (2u^2 sin 2\theta) / g^2 = (2/g) R\). Thus, \(T_1 T_2 \propto R\).