How to measure g?

Classical Mechanics Level 3

The figure shows a method for measuring the acceleration due to gravity. The ball is projected upwards by a "gun" . The ball passes the electronic" gates"1 and 2 as it rises and again as it falls . Each gate is connected to a separate timer . The first passage of the ball through each gate starts the corresponding timer ,and the second passage through the same gate stops the timer . The time intervals t 1 t_{1} and t 2 t_{2} are thus measured . The vertical distance between the two gates is "d". If d=5m and t 1 t_{1} =3 s and t 2 t_{2} =2 s , the find the measured value of acceleration due to gravity in S I. Units.

9.8m s − 2 s^{-2} 4m s − 2 s^{-2} 2m s − 2 s^{-2} 8m s − 2 s^{-2}

Prashant Kr by Prashant Kr , 22, India

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1 solution

Garrett Clarke
Jun 21, 2015

Let's denote x x as the distance from the peak of the ball to the gate. The initial velocity when the ball is at its peak is equal to 0 0 . The time taken from the peak to each of the respect gates should be be half the recorded values.

Using the equation d = v i t + 1 2 a t 2 d=v_i t+\frac{1}{2}at^2 , we make two equations by plugging in the values we know:

x = ( 0 m / s ) ( 2 s / 2 ) + 1 2 a ( 2 s / 2 ) 2 x=(0m/s)(2s/2)+\frac{1}{2}a(2s/2)^2

x + 5 m = ( 0 m / s ) ( 3 s / 2 ) + 1 2 a ( 3 s / 2 ) 2 x+5m=(0m/s)(3s/2)+\frac{1}{2}a(3s/2)^2

Now, solving for a a :

x = 1 2 a x=\frac{1}{2}a

1 2 a + 5 = 1 2 a 9 4 \frac{1}{2}a+5=\frac{1}{2}a\frac{9}{4}

5 = 5 8 a 5=\frac{5}{8}a

a = 8 m / s 2 a=\boxed{8m/s^2}

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