Acceleration calculations

Calculating acceleration An acceleration can be calculated if we know the change in velocity, and the time taken for the change to happen. This can be written as where: acceleration= change in velocity time taken a = v−u t a is the acceleration v is the final velocity u is the initial velocity t is the time taken for the change in velocity.

Calculating acceleration Acceleration can also be calculated from the change in velocity and the distance travelled while the velocity was changing. (final velocity)2 – (initial velocity)2 = 2 × acceleration × distance This can be written as v2 – u2 = 2 × a × x where: a is the acceleration v is the final velocity u is the initial velocity x is the distance travelled while the velocity was changing.

Any text in blue is a hint to help you through the question. Questions 1 Which of these formulae do you need to use to answer each of these questions? a A car is travelling at 5 m/s. It accelerates to 15 m/s in 2 seconds. What is its acceleration? Which quantities are written in the question? initial velocity, final velocity, time, acceleration Which formula has these quantities in it? a = v−u t v2 – u2 = 2 × a × x Any text in blue is a hint to help you through the question. a = v−u t

Questions 1 Which of these formulae do you need to use to answer each of these questions? b The brakes on a lorry provide a deceleration of 2 m/s2. How long does it take the lorry to slow down from 25 m/s to 17 m/s? Which quantities are written in the question? acceleration, time, initial velocity, final velocity Which formula has these quantities in it? a = v−u t v2 – u2 = 2 × a × x a = v−u t

Questions 1 Which of these formulae do you need to use to answer each of these questions? c A car travels 500 metres while accelerating from 5 m/s to 10 m/s. What is its acceleration? Which quantities are written in the question? distance, initial velocity, final velocity, acceleration Which formula has these quantities in it? v2 – u2 = 2 × a × x a = v−u t v2 – u2 = 2 × a × x

Questions 2 A car is travelling at 5 m/s. It accelerates to 15 m/s in 2 seconds. Calculate the acceleration. Write the formula: Substitute the numbers: Calculate: a = 5 m/s2 a = v−u t Make sure you write the two velocities in the correct order. The starting velocity is always last. a = (15 m/s−5 m/s) 2 s a = 10 m/s 2 s

If the object is slowing down, the acceleration will be negative. Questions 3 A cyclist is travelling at 10 m/s and applies the brakes. She slows down to 2 m/s in 4 seconds. Calculate her acceleration. Write the formula: Substitute the numbers: Calculate: a = v−u t Make sure you write the two velocities in the correct order. The starting velocity is always last. a = (2 m/s−10 m/s) 4 s If the object is slowing down, the acceleration will be negative. a = −8 m/s 4 s a = −2 m/s2

Questions 4 The brakes on a lorry provide a deceleration of 2 m/s2. How long does it take the lorry to slow down from 25 m/s to 17 m/s? Write the formula: Rearrange the formula to make t the subject: a = v−u t t = v−u a Continued

Questions 4 The brakes on a lorry provide a deceleration of 2 m/s2. How long does it take the lorry to slow down from 25 m/s to 17 m/s? Substitute the numbers: Calculate: t = 4 s It is slowing down to 17 m/s, so this is the final velocity. t = (17 m/s−25 m/s) −2 m/s2 A deceleration (slowing down) is a negative acceleration. a = −8 m/s −2 m/s2

Questions 5 A motorcycle is travelling at 10 m/s. It accelerates at 6 m/s2 for 4 seconds. Calculate its final velocity. Write the formula: Rearrange the formula to make the change in speed the subject: v – u = a × t Make v the subject of the formula: v = a × t + u a = v−u t + u + u Continued

Questions 5 A motorcycle is travelling at 10 m/s. It accelerates at 6 m/s2 for 4 seconds. Calculate its final velocity. Rearranged formula: v = (a × t )+ u Substitute the numbers: v = (6 m/s2 × 4 s) + 10 m/s Calculate: v = 24 m/s + 10 m/s = 34 m/s

Questions 6 A car decelerates at 3 m/s2 for 3 seconds. Its final velocity is 10 m/s. What was its initial velocity? Write the formula: Rearrange the formula to make the change in speed the subject: v – u = a × t Make u the subject of the formula: v = a × t + u v – a × t = u a = v−u t + u + u (−a × t) (−a × t) Continued

Questions 6 A car decelerates at 3 m/s2 for 3 seconds. Its final velocity is 10 m/s. What was its initial velocity? Rearranged formula: v – a × t = u Substitute the numbers: 10 m/s – (–3 m/s2) × 3 s = u Calculate: 10 m/s + 9 m/s = u 19 m/s = u u = 19 m/s

Questions 7 A car travels 50 metres while accelerating from 5 m/s to 10 m/s. What is its acceleration? Write the formula: Rearrange to make acceleration the subject: v2 – u2 = 2 × a × x Substitute the numbers: v2 – u2 = 2 × a × x (÷ 2 × x) (÷ 2 × x) (v2− u2) 2 × x = a (10 m/s × 10 m/s) – (5 m/s × 5 m/s) 2 × 50 m = a Continued

Questions 7 A car travels 50 metres while accelerating from 5 m/s to 10 m/s. What is its acceleration? Calculate 100 − 25 = a 100 0.75 = a a = 0.75 m/s2

Questions 8 A skydiver accelerates from 0 m/s to 5 m/s at an acceleration of 10 m/s2. How far do they fall during this acceleration? Write the formula: Rearrange to make distance the subject: v2 – u2 = 2 × a × x Substitute the numbers: v2 – u2 = 2 × a × x (÷ 2 × a) (÷ 2 × a) v2− u2 2 × a = x (5 m/s × 5 m/s) – (0 m/s × 0 m/s) 2 × 10 ms2 = x Continued

Questions 8 A skydiver accelerates from 0 m/s to 5 m/s at an acceleration of 10 m/s2. How far do they fall during this acceleration? Calculate 25 − 0 = x 20 1.25 = x x = 1.25 m

Questions 9 A car brakes from a speed of 30 m/s with a deceleration of 4 m/s2. It covers 100 metres while it is braking. Calculate its final speed. Write the formula: Rearrange to make distance the subject: v2 − u2 = 2 × a × x v2 = 2 × a × x + u2 Substitute the numbers: v2 = 2 × −4 m/s2 × 100 m + 30 m/s × 30 m/s v2 – u2 = 2 × a × x + u2 + u2 Continued

Questions 9 A car brakes from a speed of 30 m/s with a deceleration of 4 m/s2. It covers 100 metres while it is braking. Calculate its final speed. Substitute the numbers: v2 = 2 × −4 m/s2 × 100 m + 30 m/s × 30 m/s Calculate: v2 = −800 + 900 = 100 Find square root: v = 10 m/s