A car accelerates to a velocity of 80 m/s in 15 s, by which time it had traveled 450 m. What were its average and initial velocities?


To find the average velocity (\(v_{\text{avg}}\)) and the initial velocity (\(v_0\)) of the car, we can use the following kinematic equations:

1. **Average velocity formula:**
   \[
   v_{\text{avg}} = \frac{\text{total distance}}{\text{total time}}
   \]
   where:
   - total distance = 450 m,
   - total time = 15 s.

2. **Displacement equation for uniformly accelerated motion:**
   \[
   d = v_0 t + \frac{1}{2} a t^2
   \]
   where:
   - \(d\) is the displacement (450 m),
   - \(v_0\) is the initial velocity (unknown),
   - \(t\) is the time (15 s),
   - \(a\) is the acceleration (which we'll need to find).

3. **Final velocity formula:**
   \[
   v_f = v_0 + a t
   \]
   where:
   - \(v_f\) is the final velocity (80 m/s),
   - \(a\) is the acceleration,
   - \(t\) is the time (15 s).

### Step 1: Find the average velocity
Using the average velocity formula:
\[
v_{\text{avg}} = \frac{450}{15} = 30 \, \text{m/s}
\]

### Step 2: Find the acceleration
We can rearrange the final velocity formula to solve for acceleration (\(a\)):
\[
a = \frac{v_f - v_0}{t}
\]
But we first need to solve for \(v_0\) using the displacement equation.

### Step 3: Solve for initial velocity \(v_0\)
Using the displacement equation:
\[
450 = v_0 \times 15 + \frac{1}{2} a \times 15^2
\]

Now, using the final velocity equation \(v_f = v_0 + a t\) to substitute \(a = \frac{80 - v_0}{15}\) into the displacement equation:
\[
450 = 15v_0 + \frac{1}{2} \times \frac{80 - v_0}{15} \times 15^2
\]
Simplifying:
\[
450 = 15v_0 + \frac{1}{2} \times (80 - v_0) \times 15
\]
\[
450 = 15v_0 + 7.5 \times (80 - v_0)
\]
\[
450 = 15v_0 + 600 - 7.5v_0
\]
\[
450 = 7.5v_0 + 600
\]
\[
450 - 600 = 7.5v_0
\]
\[
-150 = 7.5v_0
\]
\[
v_0 = \frac{-150}{7.5} = -20 \, \text{m/s}
\]

Thus, the initial velocity \(v_0 = -20 \, \text{m/s}\), meaning the car started in reverse and then accelerated forward.

### Summary:
- **Average velocity**: \(30 \, \text{m/s}\)
- **Initial velocity**: \(-20 \, \text{m/s}\)

Would you like more details, or do you have any questions?

### Relative Questions:
1. What is the formula for average velocity in uniformly accelerated motion?
2. How can we calculate displacement when initial velocity and acceleration are known?
3. How does the sign of velocity indicate the direction of motion?
4. What is the role of acceleration in changing the velocity of an object?
5. How does uniform acceleration differ from variable acceleration?

### Tip:
In kinematics problems, keep track of signs for velocity and acceleration carefully, as they indicate direction of motion.

A car accelerates to a velocity of 80 m/s in 15 s, by which time it had traveled 450 m. What were its average and initial velocities?

Learn how to calculate the average and initial velocities of a car that accelerates to 80 m/s in 15 seconds while covering 450 meters. This problem explores key kinematics concepts and utilizes displacement and velocity equations. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation