A body moving with uniform acceleration Travels 84 metre in the first 6 second and 180 m in the next 5 seconds. find: (a) the initial velocity ,and (b) the acceleration of the body ?

Question

A body moving with uniform acceleration Travels 84 metre in the first 6 second and 180 m in the next 5 seconds. find: (a)  the initial velocity ,and (b) the acceleration of the body ?

Leo Abeyta · Accepted Answer

#"a) initial velocity "v_0 = 2 m/s#
#"b) acceleration "a = 4 m/s^2# #x = v_0 * t + a * t^2 / 2# #"Here we have"# #84 = 6 v_0 + 18 a# #84 + 180 = 264 = 11 v_0 + 60.5 a# #"This is a linear system of two equations in the variables "v_0# #"and "a.# #=> 14 = v_0 + 3 a" (Equation 1 divided by 6)"# #=> v_0 = 14 - 3 a# #=> 264 = 154 - 33 a + 60.5 a" (Eq. 2 with "v_0" replaced)"# #=> 110 = 27.5 a# #=> a = 110 / 27.5# #=> a = 4# #=> v_0 = 14 - 3*4 = 2#

Leo Abeyta · Answer

undefined To find the initial velocity and acceleration of the body, we can use the equations of motion.

Given that the body moves with uniform acceleration, we can use the equation:

$$ s = ut + \frac{1}{2}at^2 $$

where:
- $ s $ is the distance traveled,
- $ u $ is the initial velocity,
- $ a $ is the acceleration, and
- $ t $ is the time taken.

We have two sets of dataTo find the initial velocity and acceleration of the body, we can use the equations of motion.

Given that the body moves with uniform acceleration, we can use the equation:

$$ s = ut + \frac{1}{2}at^2 $$

where:
- $ s $ is the distance traveled,
- $ u $ is the initial velocity,
- $ a $ is the acceleration, and
- $ t $ is the time taken.

We have two sets of

1. For the first 6 seconds: $ s = 84 $ m and $ t = 6 $ s
2. For the next 5 seconds: $ s = 180 $ m and $ t = 5 $ s

Using the first set of dataTo find the initial velocity and acceleration of the body, we can use the equations of motion.

Given that the body moves with uniform acceleration, we can use the equation:

$$ s = ut + \frac{1}{2}at^2 $$

where:
- $ s $ is the distance traveled,
- $ u $ is the initial velocity,
- $ a $ is the acceleration, and
- $ t $ is the time taken.

We have two sets of

1. For the first 6 seconds: $ s = 84 $ m and $ t = 6 $ s
2. For the next 5 seconds: $ s = 180 $ m and $ t = 5 $ s

Using the first set of 
$$ 84 = 6u + \frac{1}{2} \cdot a \cdot (6)^2 $$

Using the second set of dataTo find the initial velocity and acceleration of the body, we can use the equations of motion.

Given that the body moves with uniform acceleration, we can use the equation:

$$ s = ut + \frac{1}{2}at^2 $$

where:
- $ s $ is the distance traveled,
- $ u $ is the initial velocity,
- $ a $ is the acceleration, and
- $ t $ is the time taken.

We have two sets of

1. For the first 6 seconds: $ s = 84 $ m and $ t = 6 $ s
2. For the next 5 seconds: $ s = 180 $ m and $ t = 5 $ s

Using the first set of 
$$ 84 = 6u + \frac{1}{2} \cdot a \cdot (6)^2 $$

Using the second set of 
$$ 180 = 6u + \frac{1}{2} \cdot a \cdot (5)^2 $$

Now, we have a system of two equations with two unknowns (initial velocity $ u $ and acceleration $ a $). We can solve this system of equations to find the values of $ u $ and $ a $.

HIX Tutor · Answer

undefined To find the initial velocity and acceleration of the body, we can use the formulas of motion.

Step 1: First, let's find the acceleration. We know that acceleration is the change in velocity over time. Since the body is moving with uniform acceleration, we can use the equation:

Acceleration (a) = Change in velocity / Time

In the first 6 seconds, the body traveled 84 meters. So, we can find the velocity at the end of the 6 seconds using the equation:

Velocity (v₁) = Initial velocity (u) + acceleration (a) * time

We can rearrange this equation to find acceleration (a):

a = (v₁ - u) / t

Substituting the values, we get:

a = (84 - u) / 6

Step 2: Next, let's find the initial velocity (u). We can use the equation:

Distance (s) = Initial velocity (u) * time + 0.5 * acceleration (a) * time²

In the next 5 seconds, the body traveled 180 meters. Substituting the values, we get:

180 = u * 5 + 0.5 * a * 5²

Simplifying this equation gives us:

180 = 5u + 12.5a

Step 3: Now, we have two equations:

a = (84 - u) / 6

180 = 5u + 12.5a

We can solve these equations simultaneously to find the values of u and a.

First, let's take the value of a from the first equation and substitute it into the second equation:

180 = 5u + 12.5((84 - u) / 6)

Simplifying this equation gives us:

180 = 5u + 14u - (5u²)/6

Rearranging, we have:

5u² - (84 * 6 - 180 * 6) = 0

This is a quadratic equation. Solving it gives us two possible values for u.

Step 4: To find the acceleration (a), substitute the value of u back into the first equation:

a = (84 - u) / 6

Repeat the steps with the second possible value for u to find the second possible value for a.

By following these steps, you will be able to find the initial velocity (u) and acceleration (a) of the body.

HIX Tutor · Answer

undefined To find the initial velocity and acceleration of the body, we can use the equations of motion.

(a) To find the initial velocity:
We know that the body traveled 84 meters in the first 6 seconds. Using the equation of motion, s = ut + (1/2)at^2, where s is the distance, u is the initial velocity, t is the time, and a is the acceleration.
Plugging in the values, we have 84 = u * 6 + (1/2) * a * 6^2.
Simplifying this equation, we get 84 = 6u + 18a.

(b) To find the acceleration:
We also know that the body traveled 180 meters in the next 5 seconds. Using the same equation of motion, s = ut + (1/2)at^2, where s is the distance, u is the initial velocity, t is the time, and a is the acceleration.
Plugging in the values, we have 180 = u * 5 + (1/2) * a * 5^2.
Simplifying this equation, we get 180 = 5u + 12.5a.

We now have two equations:
84 = 6u + 18a
180 = 5u + 12.5a

We can solve these equations simultaneously to find u and a.

Step 1: Multiply the first equation by 5 and the second equation by 6:
420 = 30u + 90a
1080 = 30u + 75a

Step 2: Subtract the first equation from the second equation:
1080 - 420 = (30u + 75a) - (30u + 90a)
660 = -15a

Step 3: Solve for a:
Divide both sides by -15:
660 / -15 = a

Step 4: Calculate a:
a = -44 m/s^2

Step 5: Substitute the value of a into the first equation to solve for u:
84 = 6u + 18a
84 = 6u + 18(-44)
84 = 6u - 792

Step 6: Solve for u:
Subtract 6u from both sides:
792 = -6u

Divide both sides by -6:
792 / -6 = u

Step 7: Calculate u:
u = -132 m/s

So, the initial velocity is -132 m/s and the acceleration is -44 m/s^2.