Question

a 2000 kilogram sports car accelerates at a rate of 30 meters per second squared. the velocity of the car is ( v = at ), where ( a ) is acceleration in meters per second squared, and ( t ) is time in seconds. the kinetic energy of the car is ( ke = \frac{1}{2}mv^2 ), where ( m ) is the mass in kilograms. which function represents the kinetic energy of the car after accelerating for ( t ) seconds? ( circ ke = 1000t^2 ) ( circ ke = 30,000t^2 ) ( circ ke = 60,000t^2 ) ( circ ke = 900,000t^2 )

Explanation:

Step1: Substitute \( v = at \) into \( KE=\frac{1}{2}mv^2 \)

Given \( a = 30 \, \text{m/s}^2 \), \( m = 2000 \, \text{kg} \), and \( v=at \). Substitute \( v \) into the kinetic energy formula:
\( KE=\frac{1}{2}m(at)^2 \)

Step2: Plug in \( m = 2000 \) and \( a = 30 \)

Substitute \( m = 2000 \) and \( a = 30 \) into the equation:
\( KE=\frac{1}{2}\times2000\times(30t)^2 \)

Step3: Simplify the expression

First, calculate \( (30t)^2 = 900t^2 \). Then:
\( KE=\frac{1}{2}\times2000\times900t^2 \)
\( \frac{1}{2}\times2000 = 1000 \), so:
\( KE = 1000\times900t^2 = 900000t^2 \)? Wait, no—wait, let's recalculate:

Wait, \( (30t)^2 = 30^2 \times t^2 = 900t^2 \). Then \( \frac{1}{2} \times 2000 = 1000 \). Then \( 1000 \times 900t^2 = 900000t^2 \)? But wait, the options have 900,000? Wait, the original problem says "2000 kilogram" (I think the first word is "A 2000-kilogram..."). Let's check again:

Wait, \( KE=\frac{1}{2}mv^2 \), \( v = at = 30t \). So \( v^2 = 900t^2 \). \( m = 2000 \). So \( \frac{1}{2} \times 2000 \times 900t^2 = 1000 \times 900t^2 = 900000t^2 \). Wait, but the options: one of them is \( KE = 900,000t^2 \)? Wait, the last option is \( KE = 900,000t^2 \)? Let me check the options again:

Options:

• \( KE = 1000t^2 \)
• \( KE = 30,000t^2 \)
• \( KE = 60,000t^2 \)
• \( KE = 900,000t^2 \)

Wait, maybe I made a mistake. Wait, \( a = 30 \), \( m = 2000 \). Let's recalculate:

\( KE = \frac{1}{2} m (at)^2 = \frac{1}{2} \times 2000 \times (30t)^2 \)
\( = 1000 \times 900 t^2 \)
\( = 900000 t^2 \)

Yes, so the correct function is \( KE = 900,000t^2 \).

Answer:

\( KE = 900,000t^2 \) (the last option, e.g., if the last option is D. \( KE = 900,000t^2 \), then D. \( KE = 900,000t^2 \))