James practises using the stopwatch facility on his new watch by measuring the time between lamp posts on a car journey. As the car speeds up, two consecutive times are 1.2 s and 1 s. Later he finds out that the lamp posts are 30 m apart.
i) Calculate the acceleration of the car (assumed constant) and its speed at the first lamp post.
ii) Assuming the same acceleration, find the time the car took to travel the 30 m before the first lamp post.

Question

Accepted Answer

i) The diagram shows all the information assuming the acceleration is a [latex]ms^{-2}[/latex] and the velocity at A is u [latex]ms^{-1}[/latex].

For AB, s = 30 and t = 1.2. You are using u and you want a so you use

[latex]s = ut + \frac{1}{2}at^{2}[/latex]

30 = 1.2u + [latex] \frac{1}{2}a × 1.2^{2}[/latex]

30 = 1.2u + 0.72a ①

To use the same equation for the part BC you would need the velocity at B and this brings in another unknown. It is much better to go back to the beginning and consider the whole of AC with s = 60 and t = 2.2. Then again using [latex] s = ut + \frac{1}{2}at^{2}[/latex]

[latex]60 = 2.2u + \frac{1}{2}a × 2.2^{2}[/latex]

60 = 2.2u + 2.42a ②

These two simultaneous equations in two unknowns can be solved more easily if they are simplified. First make the coefficients of u integers.

① × 10 ÷ 12 25 = u + 0.6a ③

② × 5 300 = 11u + 12.1a ④

then ③ × 11 275 = 11u + 6.6a ⑤

Subtracting gives

25 = 0 + 5.5a

a = 4.545

Now substitute 4.545 for a in ③ to find

u = 25 - 0.6 × 4.545 = 22.273 .

The acceleration of the car is 4.55 [latex]ms^{-2}[/latex] and the initial speed is 22.3 [latex]ms{-1}[/latex] (correct to 3 s.f.).

For this part, you know that s = 30, v = 22.3 and a = 4.55 and you want t so you use the fifth formula.

[latex]s = vt - \frac{1}{2}at^{2}[/latex]

[latex]30 = 22.3 × t - \frac{1}{2} × 4.55 × t^{2}[/latex]

⇒ 2.275t² - 22.3t + 30 = 0

Solving this using the quadratic formula gives t = 1.61 and t = 8.19.

The most sensible answer to this particular problem is 1.61 s.

Question 1.7: James practises using the stopwatch facility on his new watc...

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