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A roller coaster is moving at
25m//s at the bottom of a hill. Three seconds later it reaches the top of the hill moving at
10m//s. What was the acceleration of the coaster?

A roller coaster is moving at $25 \mathrm{~m} / \mathrm{s}$ at the bottom of a hill. Three seconds later it reaches the top of the hill moving at $10 \mathrm{~m} / \mathrm{s}$ . What was the acceleration of the coaster?

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Relate position, velocity, speed, and acceleration using derivatives

Full solution

Q. A roller coaster is moving at

25 \mathrm{~m} / \mathrm{s}

at the bottom of a hill. Three seconds later it reaches the top of the hill moving at

10 \mathrm{~m} / \mathrm{s}

. What was the acceleration of the coaster?

Identify Initial Parameters: Identify the initial velocity ( $v_i$ ), final velocity ( $v_f$ ), and the time ( $t$ ) it takes to change from the initial to the final velocity. $\newline$ Initial velocity ( $v_i$ ) = $25$ m/s (at the bottom of the hill) $\newline$ Final velocity ( $v_f$ ) = $10$ m/s (at the top of the hill) $\newline$ Time ( $t$ ) = $3$ seconds
Calculate Acceleration Formula: Use the formula for acceleration $a$ , which is the change in velocity $\Delta v$ divided by the time $t$ it takes for the change. $\newline$ Acceleration $a$ = (Final velocity - Initial velocity) / Time $\newline$ Acceleration $a$ = $\frac{v_f - v_i}{t}$
Plug in Values: Plug in the known values into the acceleration formula to calculate the acceleration. $\newline$ Acceleration $a$ = $\frac{10 \, \text{m/s} - 25 \, \text{m/s}}{3 \, \text{s}}$ $\newline$ Acceleration $a$ = $\frac{-15 \, \text{m/s}}{3 \, \text{s}}$
Calculate Acceleration: Calculate the acceleration. $\newline$ Acceleration $a = -5 \, \text{m/s}^2$ $\newline$ The negative sign indicates that the roller coaster is decelerating as it climbs the hill.

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