The rocket-driven sled Sonic Wind No. 2, used for investigating the physiological effects of large accelerations, runs on a straight, level track 1070 m (3500 ft) long. Starting from rest, it can reach a speed of 224 m/s (500 mi/h) in 0.900 s. (a) Compute the acceleration in m/s2, assuming that it is constant. (b) What is the ratio of this acceleration to that of a freely falling body (g) ? (c) What distance is covered in 0.900 s? (d) A magazine article states that at the end of a certain run, the speed of the sled de-creased from 283 m/s (632 mi/h) to zero in 1.40 s and that during this time the magnitude of the acceleration was greater than 40 g. Are these figures consistent?

a) The acceleration of the rocket is 249 m/s².

b) The acceleration of the rocket is 25 times the acceleration of a free-falling body (25 g),

c) The distance traveled in 0.900 s was 101 m.

d) The figures are not consistent. The acceleration of the rocket was 20 g.

Explanation:

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a) To calculate the acceleration of the rocket let's use the equation of velocity of the rocket:

v = v0 + a · t

Where:

v = velocity of the rocket.

v0 = initial velocity.

a = acceleration.

t = time.

We know that at t = 0.900 s, v = 224 m/s. The initial velocity, v0, is zero because the rocket starts from rest.

v = v0 + a · t

Solving for a:

(v - v0) / t = a

224 m/s / 0.900 s = a

a = 249 m/s²

The acceleration of the rocket is 249 m/s²

b) The acceleration of gravity is ≅ 10 m/s². The ratio of the acceleration of the rocket to the acceleration of gravity will be:

249 m/s² / 10 m/s² = 25

So, the acceleration of the rocket is 25 times the acceleration of gravity or 25 g.

c) The equation of traveled distance is the following:

x = x0 + v0 · t + 1/2 · a · t²

Where:

x = position of the rocket at time t.

x0 = initial position.

v0 = initial velocity.

t = time.

a = acceleration.

Since x0 and v0 are equal to zero, then, the equation of position gets reduced to:

x = 1/2 · a · t²

x = 1/2 · 249 m/s² · (0.900 s) ²

x = 101 m

The distance traveled in 0.900 s was 101 m.

d) Now, using the equation of velocity, let's calculate the acceleration. We know that at 1.40 s the velocity of the rocket is zero and that the initial velocity is 283 m/s.

v = v0 + a · t

0 m/s = 283 m/s + a · 1.40 s

-283 m/s / 1.40 s = a

a = - 202 m/s²

The figures are not consistent because 40 g is equal to an acceleration of 400 m/s² and the magnitude of the acceleration of the rocket was ≅20 g.