A truck covers 50.0 m in 9.50 s while smoothly slowing down to a final speed of 2.80
m/s. (a) Find the truck’s original speed. (b) Find its acceleration.

To solve the problem, we can use the kinematic equations of motion, which relate displacement, initial velocity, final velocity, acceleration, and time.

### Given:
- Displacement, \( s = 50.0 \, \text{m} \)
- Time, \( t = 9.50 \, \text{s} \)
- Final velocity, \( v_f = 2.80 \, \text{m/s} \)

### (a) Finding the Truck’s Original Speed (Initial Velocity \(v_i\))

We can use the kinematic equation:
\[
s = v_i t + \frac{1}{2} a t^2
\]
We also know that:
\[
v_f = v_i + a t
\]

To find \( v_i \), we need to eliminate \( a \) from the equation. First, solve the second equation for \( a \):
\[
a = \frac{v_f - v_i}{t}
\]
Now, substitute this expression for \( a \) into the first equation:
\[
s = v_i t + \frac{1}{2} \left(\frac{v_f - v_i}{t}\right) t^2
\]
Simplify to:
\[
s = v_i t + \frac{1}{2} (v_f - v_i) t
\]
\[
s = \frac{v_i t + v_f t - v_i t}{2}
\]
\[
s = \frac{v_f t + v_i t}{2}
\]
\[
2s = (v_i + v_f) t
\]
\[
v_i = \frac{2s}{t} - v_f
\]
Substitute the known values:
\[
v_i = \frac{2 \times 50.0 \, \text{m}}{9.50 \, \text{s}} - 2.80 \, \text{m/s}
\]
\[
v_i = \frac{100.0 \, \text{m}}{9.50 \, \text{s}} - 2.80 \, \text{m/s}
\]
\[
v_i \approx 10.53 \, \text{m/s} - 2.80 \, \text{m/s}
\]
\[
v_i \approx 7.73 \, \text{m/s}
\]

### (b) Finding the Acceleration \(a\)

Now that we have \( v_i \), we can find the acceleration using:
\[
a = \frac{v_f - v_i}{t}
\]
\[
a = \frac{2.80 \, \text{m/s} - 7.73 \, \text{m/s}}{9.50 \, \text{s}}
\]
\[
a \approx \frac{-4.93 \, \text{m/s}}{9.50 \, \text{s}}
\]
\[
a \approx -0.519 \, \text{m/s}^2
\]

### Final Answers:
(a) The truck’s original speed is approximately \( 7.73 \, \text{m/s} \).

(b) The acceleration is approximately \( -0.519 \, \text{m/s}^2 \), indicating that the truck is decelerating.

Would you like more details or have any questions?

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Here are some related questions:
1. How would the initial speed change if the final speed were higher?
2. What would be the acceleration if the truck came to a complete stop?
3. How does the time taken to cover a certain distance affect the initial speed calculation?
4. If the truck had an initial speed of 10 m/s, how would the final speed change for the same deceleration?
5. How would the displacement change if the truck maintained a constant speed?

**Tip:** Always double-check the units in kinematic equations to ensure consistency and correctness.

Learn how to find the original speed and acceleration of a truck using kinematic equations. Given initial displacement, final velocity, and time, calculate the truck’s speed and acceleration. Detailed step-by-step solution included.

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