An object starts with a speed of u m/s and travels along a straight line. Speed increases at the rate of a m/s every second. The distance travelled in time t seconds is ut + 1/2 at^2. The distance travelled in 2 seconds is 10 metres and the distance travelled in 4 seconds is 28 metres. What was the initial speed? What is the rate at which speed is increasing?
Answer (1)
To solve this problem, we need to find two unknowns: the initial speed u in meters per second (m/s) and the acceleration a in meters per second squared (m/s²). We are given that the distance traveled in time t is given by the equation:
s = u t + 2 1 a t 2
We have two conditions to work with:
The distance traveled in 2 seconds is 10 meters. 10 = 2 u + 2 1 a ( 2 2 ) 10 = 2 u + 2 a
The distance traveled in 4 seconds is 28 meters. 28 = 4 u + 2 1 a ( 4 2 ) 28 = 4 u + 8 a
Now, we have a system of two linear equations:
Equation 1: 2 u + 2 a = 10
Equation 2: 4 u + 8 a = 28
Let's simplify these equations:
For Equation 1: u + a = 5 (Dividing the entire equation by 2)
For Equation 2: 2 u + 4 a = 14 (Dividing the entire equation by 2)
Now, we can solve these equations simultaneously.
From the first equation: u = 5 − a
Substitute u = 5 − a into the second simplified equation: 2 ( 5 − a ) + 4 a = 14 10 − 2 a + 4 a = 14 10 + 2 a = 14 2 a = 4 a = 2
With a = 2 , substitute back to find u : u = 5 − a u = 5 − 2 u = 3
So, the initial speed u is 3 m/s and the acceleration a is 2 m/s².
