Motion in a Straight Line — JEE Mains Physics

Formulas (2)

• Average speed over a journey in legs · Average speed
  $$\bar v = \frac{\text{total distance}}{\text{total time}} \qquad \text{equal distances: } \bar v = \frac{2v_1v_2}{v_1 + v_2}$$
• Relative velocity in one dimension · Relative velocity
  $$v_{BA} = v_B - v_A \qquad t_{cross} = \frac{L_1 + L_2}{v_{rel}}$$

Watch out for (6)

• Averaging two speeds over equal distances
  → Average speed over a journey in legs
• Average velocity on a round trip is zero
  → Average speed over a journey in legs
• Equal times inside an equal-distance leg
  → Average speed over a journey in legs
• Forgetting the train's own length
  → Relative velocity in one dimension
• Subtracting speeds for trains approaching each other
  → Relative velocity in one dimension
• Leaving speeds in km/h
  → Relative velocity in one dimension

Formulas (3)

• Choosing the right equation of motion · Equations of motion
  $$v = u + at \qquad s = ut + \tfrac{1}{2}at^{2} \qquad v^{2} = u^{2} + 2as \qquad s = \frac{u + v}{2}\,t$$
• Braking and stopping distance · Stopping distance
  $$s = \frac{u^{2}}{2a} \qquad \frac{s_2}{s_1} = \left(\frac{u_2}{u_1}\right)^{2}$$
• Distance in the nth second · Distance in the nth second
  $$s_n = u + \frac{a}{2}(2n - 1)$$

Watch out for (6)

• Mid-distance speed is not the mean speed
  → Choosing the right equation of motion
• The ratio of times is upside down
  → Choosing the right equation of motion
• Stopping distance is not proportional to speed
  → Braking and stopping distance
• Loses one third, or keeps one third?
  → Braking and stopping distance
• "In the nth second" is not "in n seconds"
  → Distance in the nth second
• Seconds two apart differ by 2a
  → Distance in the nth second

Formulas (3)

• Free fall from rest · Falling from rest
  $$h = \tfrac{1}{2}gt^{2} \qquad v = gt \qquad v = \sqrt{2gh}$$
• Thrown up or down: one equation for the whole flight · Whole flight, up positive
  $$s = ut - \tfrac{1}{2}gt^{2} \qquad H = \frac{u^{2}}{2g} \qquad t_{drop} = \sqrt{t_1t_2}$$
• Two bodies, or drops falling at regular intervals · Later body's position
  $$y_2 = u_2(t - \tau) - \tfrac{1}{2}g(t - \tau)^{2}$$

Watch out for (7)

• A rebound changes the sign of the velocity
  → Free fall from rest
• Using 9.8 when the stem gives 10
  → Free fall from rest
• Dropped from a moving body is not dropped from rest
  → Thrown up or down: one equation for the whole flight
• Zero velocity is not zero acceleration
  → Thrown up or down: one equation for the whole flight
• Splitting the flight when one equation will do
  → Thrown up or down: one equation for the whole flight
• Counting drops instead of intervals
  → Two bodies, or drops falling at regular intervals
• Giving both bodies the same clock
  → Two bodies, or drops falling at regular intervals

Formulas (2)

• Areas under velocity–time and acceleration–time graphs · Areas
  $$\Delta x = \int v\,dt \qquad \Delta v = \int a\,dt$$
• Velocity–position graphs: a = v dv/dx · Acceleration from velocity and position
  $$a = v\frac{dv}{dx} = \frac{1}{2}\frac{d(v^{2})}{dx}$$

Reference tables (1)

Watch out for (6)

• Area gives displacement, not distance
  → Areas under velocity–time and acceleration–time graphs
• Average velocity comes from areas on a v–t graph
  → Areas under velocity–time and acceleration–time graphs
• Average velocity is a chord, not a tangent
  → Matching graph shapes to the motion
• Velocity zero does not make the acceleration zero
  → Matching graph shapes to the motion
• The slope of v–x is not the acceleration
  → Velocity–position graphs: a = v dv/dx
• Halving the slope of v² against x
  → Velocity–position graphs: a = v dv/dx

Formulas (3)

• Position as a function of time: differentiate · Differentiate
  $$v = \frac{dx}{dt} \qquad a = \frac{dv}{dt} = \frac{d^{2}x}{dt^{2}}$$
• Velocity as a function of position: a = v dv/dx · Chain rule
  $$a = v\frac{dv}{dx} \qquad v = k\sqrt{x} \Rightarrow a = \frac{k^{2}}{2}$$
• Acceleration or force as a function of time: integrate · Integrate with the starting values
  $$v(t) = u + \int_0^{t} a\,dt \qquad x(t) = x_0 + \int_0^{t} v\,dt$$

Watch out for (6)

• A turning point is v = 0, not a = 0
  → Position as a function of time: differentiate
• Distance across a turning point
  → Position as a function of time: differentiate
• dv/dx is not the acceleration
  → Velocity as a function of position: a = v dv/dx
• Inverting the derivative but not the function
  → Velocity as a function of position: a = v dv/dx
• Dropping the starting value
  → Acceleration or force as a function of time: integrate
• Differentiating when you should integrate
  → Acceleration or force as a function of time: integrate