We can solve this problem using the kinematic equations for motion with constant acceleration. The equation that relates displacement ($d$), initial velocity ($v_0$), time ($t$), and acceleration ($a$) is:

$d = v_0 t + \frac{1}{2} a t^2$

We also know that acceleration can be found using the following equation, given the final velocity ($v$) and the initial velocity ($v_0$):

$v = v_0 + a t$

Given:

• Final velocity ($v$) = 7.5 m/s
• Time ($t$) = 4.5 s
• Displacement ($d$) = 19 m

We have two unknowns, $v_0$ (initial velocity) and $a$ (acceleration). To find $v_0$, we'll first find $a$ using the second equation, then substitute it into the first equation.

Step 1: Find acceleration ($a$)

Rearrange the second equation to solve for $a$:

$a = \frac{v - v_0}{t}$

Substitute this into the first equation.

Step 2: Substitute $a$ into the displacement equation

$d = v_0 t + \frac{1}{2} \left(\frac{v - v_0}{t}\right) t^2$

Simplify this equation:

$d = v_0 t + \frac{1}{2} (v - v_0) t$

$d = v_0 t + \frac{1}{2} v t - \frac{1}{2} v_0 t$

$d = \left(v_0 - \frac{1}{2} v_0\right) t + \frac{1}{2} v t$

$d = \frac{1}{2} v_0 t + \frac{1}{2} v t$

Multiply both sides by 2 to eliminate the fractions:

$2d = v_0 t + v t$

Now, solve for $v_0$:

$v_0 t = 2d - v t$

$v_0 = \frac{2d - v t}{t}$

Step 3: Substitute the known values

$v_0 = \frac{2(19) - (7.5)(4.5)}{4.5}$

Let's calculate this.

$v_0 = \frac{38 - 33.75}{4.5} = \frac{4.25}{4.5}$

$v_0 \approx 0.94 \, \text{m/s}$

Conclusion:

The initial velocity of the bike was approximately 0.94 m/s.

Would you like more details or explanations?

Here are some related questions you might find helpful:

1. How do we calculate acceleration when displacement is not given?
2. What if the rider starts from rest—how would the equations change?
3. How does constant acceleration affect the total displacement?
4. Can we calculate the rider's average velocity during this period?
5. What other kinematic equations can be used for constant acceleration?

Tip: Always check if you can simplify your problem using known values before solving for unknowns in more complex equations.