JEE Physics Motion In A Straight Line Notes Set 07

Motion in a Straight Line: Quick Revision

Distance is the length of the actual path and is never negative. Displacement is the shortest straight-line change of position from start to end and carries a sign. Distance is at least as large as the magnitude of displacement, and average speed is at least as large as the magnitude of average velocity. If the two are unequal, the body must have reversed its motion, so its velocity was zero at some instant.

Average and Instantaneous Quantities

Average velocity is displacement divided by time taken; average speed is total path length divided by time taken. Instantaneous velocity is \( v=\frac{dx}{dt} \) and instantaneous acceleration is \( a=\frac{dv}{dt}=v\frac{dv}{dx} \). The second form is the one to use whenever velocity is given as a function of position.

Two averages occur again and again. If a body covers equal distances at speeds \( v_1 \) and \( v_2 \), its average speed is the harmonic mean \( \frac{2v_1v_2}{v_1+v_2} \). If it moves for equal times at \( v_1 \) and \( v_2 \), the average speed is the arithmetic mean \( \frac{v_1+v_2}{2} \).

Reading Graphs

1. The slope of a position-time graph at a point is the instantaneous velocity.
2. The slope of a velocity-time graph is the acceleration.
3. The area under a velocity-time graph is the displacement; the area under an acceleration-time graph is the change in velocity.
4. Position-time graphs cannot have sharp corners with an instantaneous jump in slope in real motion, because velocity cannot change in zero time.
5. A body moving with uniform acceleration from rest has a parabolic position-time graph, and a straight-line velocity-time graph through the origin.

Equations of Uniformly Accelerated Motion

These hold only when acceleration is constant. If acceleration varies with time or position, integrate \( a=\frac{dv}{dt} \) or \( v\,dv=a\,dx \) instead. For motion from rest, distances in successive equal time intervals are in the ratio 1 : 3 : 5 : 7, and the distance covered after time t grows as \( t^2 \).

Motion Under Gravity

Throwing a body up with speed u (air resistance ignored): maximum height \( H=\frac{u^2}{2g} \), time of ascent equals time of descent \( =\frac ug \), total flight time \( \frac{2u}{g} \), and it returns to the starting level with the same speed u. At any level the speed on the way up equals the speed on the way down.

If the body passes a point at heights h at times \( t_1 \) and \( t_2 \), then \( h=\frac12gt_1t_2 \) and the maximum height is \( \frac18g(t_1+t_2)^2 \). If bodies are thrown up, down and horizontally from the same height with the same speed and take \( t_1, t_2, t_3 \) to reach the ground, then \( t_3=\sqrt{t_1t_2} \).

Points Often Tested

1. A body can have zero velocity and still have non-zero acceleration (the top of a throw).
2. A positive acceleration may slow a body down if velocity is negative; the signs decide, not the symbol.
3. Average velocity can equal instantaneous velocity at some moment (mean value idea).
4. Two bodies released or thrown under gravity have zero relative acceleration, so their relative velocity stays constant.
5. In a lift accelerating up at a, a falling object has effective gravity \( g+a \) relative to the lift.
6. When a body is released from a moving object, it keeps the velocity of that object at the instant of release.