Motion in Straight Line PDF

# Motion in Straight Line - Distance - Actual path or total path - Scalar quantity - Always positive, cannot be negative. - Displacement - Shortest distance between two points. - Change in position = *x* - *x*₀ - Vector quantity - May be positive, negative or z...

# Motion in Straight Line - Distance - Actual path or total path - Scalar quantity - Always positive, cannot be negative. - Displacement - Shortest distance between two points. - Change in position = *x* - *x*₀ - Vector quantity - May be positive, negative or zero. - Area of velocity-time graph - Distance ≥ Displacement | Displacement | Displacement is zero | Displacement is not zero | Displacement is not zero or may not be zero | It distance is not zero | It distance must be zero | Distance must be zero | Distance may or may not be zero | |---|---|---|---|---|---|---|---| | is zero | | | | | | | | | may be zero | | | | | | | | | may not be zero | | | | | | | | | It helps in Assertion & Reasoning Questions | | | | | | | | # Distance & Displacement on Circular Path - Distance = Arc = Rθ - Displacement = 2Rsin(θ/2) - Average velocity = sin(θ/2) / (θ/2) # Important points - Instantaneous velocity = *dv*/ *dt* - Average velocity = Total displacement/Total time - Average speed = Total distance/Total time - |Average velocity| = |Average speed| (Only when direction does not change) - Instantaneous speed = |Instantaneous velocity| (Always) # Important case - An object moving in a straight line covers *x₁* distance in time *t₁*, and *x₂* distance in *t₂* time. - Average speed = (*x*₁ + *x*₂)/ (*t*₁ + *t*₂) - Unequal time intervals with different speed (*V*₁ for *t*₁, and *V*₂ for *t*₂) - Average speed = (*V*₁*t*₁ + *V*₂*t*₂) / (*t*₁ + *t*₂) - Unequal distance covers with different speed (*x*₁ with *V*₁, and *x*₂ with *V*₂) - Average speed = (*V*₁*V*₂ (*x*₁ + *x*₂) / (*V*₁*x*₂ + *V*₂*x*₁) - Equal distance cover with different speed - Average speed = 2*V*₁*V*₂ / (*V*₁ + *V*₂) - Travel half time with speed *V*₁. Another half time with speed *V*₂. - Average speed = (*V*₁ + *V*₂) / 2 # Half distance with speed *V*₁. In another half distance, half time with *V*₂ and half time with *V*₃. - Average speed = 2*V*₁(√2 + √3) / (2*V*₁ + *V*₂+ *V*₃) - Travel 3 equal distance with *V*₁, *V*₂, *V*₃ respectively. - *V_avg* = 3*V*₁*V*₂*V*₃ / (*V*₁*V*₂ + *V*₁*V*₃ + *V*₂*V*₃) # Acceleration - Rate of change of velocity. - *a*_inst = *dv*/ *dt* = *d²x*/ *dt² * - *a*_avg = Δ*v*/ Δ*t* = (*V*_f - *V*_i)/ *t* = ∫*a* *dt*/ *t* # Golden Rule - Integration = Differentiation - Integration = Differentiaion - Integration = Area under graph - Differentiation = Slope # Case 1 (Acceleration = 0) - *a* = 0 = *d*²*x*/ *dt² - Velocity = constant (uniform motion) - *V*_inst = *V*_avg - distance = *v* *t* # Case 2 (Acc = variable) - Non-uniformly accelerated motion - Then we will use differentiation and integration # Case 3 (Acc = constant) - Acceleration will be constant. - *d*² *x*/ *dt² = *a* - *v* *dt* = *a* *dx* - *v* = *v*₀ + *a* *t* # Motion with constant Acc. - Uniformly accelerated motion - *v* = *u* + *a* *t* - *s* = *u* *t* + 1/2 *a* *t*² - *v*² = *u*² + 2 *a* *s* - *V*_avg = (*u* + *v*)/ 2 - *s* = (*u* + *a* *t*) *t*/2 - Distance covered in *n*th sec. - *S*_nth = *u* + 1/2 *a* (2*n* - 1) - Transitional acc (*a_t*) = *dv*/ *dt* # Important case - Angle between *v* and *a* is 180° - Speed decrease = -ve - *a* = *a*_avg *cos* 180° - *a* = -ve - No change in direction - Angle between *v* and *a* is 0° - Speed increase - *a* = *a*_avg *cos* 0° - *a* = +ve - No change in direction - Angle between *v* and *a* is 90° - Speed constant - direction will change # Object start moving from rest with constant acceleration - u = 0, *a* = constant - 1st sec: *S*_{1st sec:} *S*_{2nd sec}: *S*_{3rd sec} = *x* : 3*x* : 5*x* - *S*_{1st sec:} *S*_{2nd sec}: *S*_{3rd sec} = *x* : 4*x* : 9*x* # Velocity at Mid Point - *V*_mid = √ *V*² + *u*² / 2 - *V*_mid = √ *V*² + *u*² / 2 # Stopping distance (S) - *S* = *u*² / 2*a* - [ : *V* = 0 ] - *S* = *u*² / 2*a* # Rest to Rest Motion - Object start from rest - Constant accm *a* for time *t*₁, and then retard with `β' & comes to rest in time *t*2' - Total time = *T* - *a* *t*<sub>1</sub> = β *t*<sub>2</sub> - *x*<sub>1</sub> = *a* *t*<sub>1</sub><sup>2</sup> / 2 - *x*<sub>2</sub> = β *t*<sub>2</sub><sup>2</sup> / 2 - *V*<sub>max</sub> = (β + *a*) *T* / (2 * √β*a*) # Ratio of time in equal distance - Valid when u = 0, acc = const. - *t*<sub>1</sub>:*t*<sub>2</sub>:*t*<sub>3</sub> = 1: (√2 - 1): (√3 - √2) - *t*<sub>1</sub>:*t*<sub>2</sub>:*t*<sub>3</sub> = 1: (√2 - 1): (√3 - √2) # Motion Under Gravity - Sign Conversion - Velocity: +ve↑ (upward motion) - Velocity: -ve↓ (downward motion) - Acceleration (g) : Always -ve - Displacement (s) : - +ve = final position is above initial position - -ve = final position is below initial position - Zero: final & initial position are at the same level. # Drop from height - *v* = *g* *t* - *v* = √2 *g* *H* - *s* = 1/2 *g* *t*<sup>2</sup> *T* = √2 *H*/ *g* - *V*<sub>avg</sub> = √*g* *H* - Time to cover half (*H*/2) =√*H*/ *g* # Ground to ground - *H*<sub>max</sub> = *u*<sup>2</sup>/ 2 *g* # Time to touch - *T*<sub>up</sub> = *T*<sub>down</sub> = *T*<sub>flight</sub> = 2 *u*/ *g* # If air resistance = `a' - *T*_up = *u*/ (*g* - *a*) - *T*_down = *u*/ (*g* + *a*) # Motion under gravity from some height (H) → - *H* = - *u* *t* + 1/2 *g* *t*² - *T* = *u* + √*u*² + 2 *H* *g* / *g* - *a* = + *g* # Graph # Position - Time graph - *x* - ---(x1,t1) - ---(x2,t2) - ----------- - **(0,0)** ----------| - ----------- - ----------- - ----------- - ----------- - ----------- - **t** - Slope = *dx*/ *dt* = velocity = *tan* θ - Velocity (at *t* = 0) → +ve and constant - Rest (V =0) - *V* = Slope = constant and +ve - Motion: start from 0 and constant velocity - V = +ve - *a* = +ve - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - **t** - V = increasing velocity - V = decreasing velocity - Retardation - --------- - -------- - -------- - -------- - -------- - -------- - Slope (V) = -ve and constant. # Velocity - Time graph - **V** - -------------------- - -------------------- - --------------------- - ----------- - ----------- - **(0,0)** -------- - -------- - -------- - -------- - -------- - -------- - -------- - -------- - -------- - -------- - **t** - Slope = *dv*/ *dt* = *a*_instant. - Area = ∫*v* *dt* = displacement - Disp = *A*₁ - *A*₂ - Distance = *A*₁ + *A*₂ - **V** - -------------------- - ---------- - ----------- - ----------- - ---------- - ----------- - --------| - --------| - --------| - --------| - --------| - --------| - --------| - --------| - --------| - --------| - **t** - *V* = const - *a* = 0 - **V** - ---------- - ----------- - ----------- - ----------- - ---------- - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - **t** - *a* < 90° - Slope = *dv*/ *dt* = *a* - Velocity increase. - **V** - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - **t** - *a* = slope = increasing - Velocity increase - **V** - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - **t** - *a* = slope = decreasing - Velocity decrease - Slope = *a* = -ve, const. - 0 > 90° # Velocity - Position graph - **V** - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - --------| - --------| - *x* - Slope = *dv*/ *dx* - *a* = *v* *dv*/ *dx* - *a* = *v* *slope* - Slope = *a*/ *v* # *v*² = *x* (graph) - **V** - ----------- - ----------- - ----------- - ----------- - ----------- - ----------- - --------| - **x** - Slope = *dv*² / *dx* = *dv*² *dv* / *dx* *dv* - Slope = 2 *V* *dv*/ *dx* - Slope = 2 *a* # *a* - *t* (graph) - **a** - ----------- - ----------- - ----------- - --------| - --------| - **t** - Area under (*a* - *t*) = ∫*a* *dt* = Δ*v* - *V*_f - *V*_i = Area of graph - *V*_f -*V*_i = *A*₁ - *A*₂ # *a* - *x* (graph) - **a** - ----------- - ----------- - ----------- - --------| - --------| - **x** - Area = 1/2 *V*_f² - *u*²/ 2 # Relative Motion in 1-D - *V*_AB = *V*_A - *V*_B - *V*_AB = - *V*_BA # Golden Rule - Same direction = Subtraction - Opposite direction = Addition # Ground - **XR** - ----> - **Xp** - ----> - **XPR** - ----> - **XRP** - ----> - *X*_R = Position of 'R' from ground - *X*_P = Position of 'P' from ground - *X*_RP = Position of 'R' w.r.t 'P' - *X*_PR = *X*_P - *X*_R - *V*_PR = *V*_P - *V*_R - *V*_PR = *V*_Pg + (-*V*_Rg) - *a*_PR = *a*_P - *a*_R # Case 1 - *V*_R = *V*_P - *V*_PR = *V*_P - *V*_R = 0 # Case 2 - *V*_R > *V*_P - *V*_PR = *V*_P - *V*_R = - ve. - *V*_RP = + ve # Case 3 - *V*_R < *V*_P - *V*_PR = *V*_P - *V*_R = + ve. - *V*_RP = - ve.